Translational, rotational, and gravitational kinematics, gravitational fields, and translational dynamics - motion, orbits, mass, and momentum.
These practice problems are introductory mechanics: how position, velocity, and acceleration are related, and how mass and force determine the motion. Calculus is used, but the question is physical. You will read a motion from its kinematic equations, treat inverse-square gravity and simple orbits, and apply momentum and Newton's laws. Each solution names the physical law before the differentiation or the integration.
Translational Kinematics covers motion on a line and in a plane, including projectiles. Rotational Kinematics is the angular analogue: angle, angular velocity, and angular acceleration. Gravitational Field and Gravitational Kinematics treat the inverse-square field and motion under gravity, including orbits. Translational Dynamics applies Newton's laws, momentum, and energy to those same motions, which is where the problems become harder.
Start with the kinematics chapters. Differential Calculus and Integrals supply the derivatives and antiderivatives the solutions use, and they are the right detour if the calculus step is the one that fails. Work and mass written as integrals also appear in Applications of Definite Integrals. The heat and wave equations are not in this set; they are in Partial Differential Equations.
Problem 1
HardTranslational Kinematics
During the first half of the total travel time, a car moves at a speed of $v_1 = 54\ \mathrm{km/h}$. During the second half of the travel time, it moves at a speed of $v_2 = 36\ \mathrm{km/h}$. What is the average speed of the car?
The speed of a car during the first half of the distance is $v_1 = 36\ \mathrm{km/h}$, and during the second half of the distance it is $v_2 = 54\ \mathrm{km/h}$. What is the average speed of the car over the entire journey?
A body moves along a straight path. Over successive segments of equal length $s$, it keeps constant speeds $v_1, v_2, v_3, \ldots, v_n$. What is the average speed of the body?
A body moves along a straight path. In successive equal time intervals of duration $\Delta t$, it keeps constant speeds $v_1, v_2, v_3, \ldots, v_n$. What is the average speed of the body?
On the first third of the total distance, a car moves at speed $v_1$. On the remaining two thirds, it moves at $v_2 = 54\ \mathrm{km/h}$. The average speed over the entire journey is $\langle v \rangle = 36\ \mathrm{km/h}$. What is the speed $v_1$?
A locomotive is moving at a speed of $v_1 = 54\ \mathrm{km/h}$. A train of length $l = 150\ \mathrm{m}$ is approaching it from the opposite direction at a speed of $v_2 = 36\ \mathrm{km/h}$. How long does it take for the train to pass the locomotive's driver completely?
One car departs from city $A$ to city $B$ every $\Delta t = 10\ \mathrm{min}$. The cars travel at a speed of $v = 60\ \mathrm{km/h}$. The distance between the cities is $d = 60\ \mathrm{km}$. a) Draw a graph of the distance travelled by a car versus time. b) At what speed should the fourth car in the column leave city $A$ so that it reaches city $B$ at the same time as the first car? When and where does this car overtake the cars ahead of it?
The distance between cities $A$ and $B$ is $d = 200\ \mathrm{km}$. At the same moment, one car leaves city $A$ and another leaves city $B$, driving toward each other. The car from $A$ has a speed of $v_A = 60\ \mathrm{km/h}$, and the car from $B$ has a speed of $v_B = 40\ \mathrm{km/h}$. a) Draw a graph of the position of each car versus time. b) Using the graph, find the time and place at which the cars meet.
A boat travels downstream a distance $s = 24\ \mathrm{km}$ with a constant speed $v_B = 8\ \mathrm{km/h}$ relative to the water. The trip takes $t = 2.5\ \mathrm{h}$. Find the speed of the river current $v_R$.
A steamboat travels between two cities on the same river. The trip upstream takes $t_1 = 9\ \mathrm{h}$, and the trip downstream takes $t_2 = 4\ \mathrm{h}$. The distance between the cities is $d = 72\ \mathrm{km}$. Find the average speed $u$ of the river current relative to the shore and the speed $v$ of the steamboat relative to the water.
A man of height $h = 1.8\ \mathrm{m}$ walks at a constant speed $v = 0.75\ \mathrm{m/s}$ away from a streetlight mounted at a height $H = 4\ \mathrm{m}$ above the ground. What is the speed of the tip of his shadow along the ground?
A theatre spectator and a radio listener hear the opening sound of an overture. (a) At what distance from the orchestra should the spectator sit so that he hears the first sound at the same time as a radio listener who is $d = 7500\ \mathrm{km}$ away from the theatre? (b) At what distance from the receiver should the radio listener be so that he hears the first sound at the same time as a spectator sitting $l_1 = 30\ \mathrm{m}$ from the orchestra? The speed of sound is $c = 340\ \mathrm{m/s}$ and the speed of radio waves (light) is $c_d = 3 \cdot 10^8\ \mathrm{m/s}$.
Two airplanes fly one behind the other along the same line, $d = 1080\ \mathrm{m}$ apart, both with speed $v_1 = 1200\ \mathrm{km/h}$ relative to the ground. A shell is fired from the rear airplane toward the front one with speed $v_2 = 300\ \mathrm{m/s}$ relative to the airplane. After what time does the shell hit the front airplane? How far does the airplane travel in this time?
An anti-tank gun fires a shell at a tank. The shell hits the tank $t_1 = 0.6\ \mathrm{s}$ after firing, and the sound of the explosion is heard at the gun $t_2 = 2.1\ \mathrm{s}$ after firing. What is the distance from the gun to the tank, and what is the average speed of the shell? The speed of sound is $c = 340\ \mathrm{m/s}$.
One train is $l_1 = 500\ \mathrm{m}$ long and another is $l_2 = 100\ \mathrm{m}$ long. They move toward each other with speeds $v_1 = 16\ \mathrm{m/s}$ and $v_2 = 20\ \mathrm{m/s}$. How long does it take for the first train to pass completely by a window of the second train? How far does the first train travel in this time?
Two airplanes fly one behind the other with the same speed, $d = 750\ \mathrm{m}$ apart. A shot is fired from the first (front) airplane, and the pilot of the second airplane hears it $t = 1\ \mathrm{s}$ later. What is the speed of the airplanes? The speed of sound is $c = 340\ \mathrm{m/s}$.
Two cities lie along a river, $d = 1\ \mathrm{km}$ apart. A boat moves with speed $v_1 = 8\ \mathrm{km/h}$ relative to the water. By opening and closing a dam, the water in one case flows with speed $v_2 = 2\ \mathrm{km/h}$, and in the other case it is still. Compare the times the boat needs to travel the route upstream and back downstream in both cases.
Under the conditions of the previous problem (Translational Kinematics 017): the cities are $d = 1\ \mathrm{km}$ apart, the boat's speed relative to the water is $v_1 = 8\ \mathrm{km/h}$, and the water either flows at $v_2 = 2\ \mathrm{km/h}$ or is still. Find the distance the boat travels relative to the water on the round trip in both cases.
A boat moves with a constant speed relative to the water between two cities on a river. The trip upstream takes $t_1 = 6\ \mathrm{h}$ and the trip downstream takes $t_2 = 3\ \mathrm{h}$. How long would the boat need to travel downstream with the engine turned off?
Two objects move along the same straight line. When they move away from each other, the distance between them changes by $\Delta d_1 = 16\ \mathrm{m}$ in $\Delta t_1 = 10\ \mathrm{s}$. When they move toward each other, the distance changes by $\Delta d_2 = 3\ \mathrm{m}$ in $\Delta t_2 = 3\ \mathrm{s}$. What are the speeds of the objects?
Two objects (1 and 2) start at the same moment from the same point and move in mutually perpendicular directions with speeds $v_1 = 30\ \mathrm{km/h}$ and $v_2 = 40\ \mathrm{km/h}$. How does the distance between them change with time? What is their distance when the first object has travelled $s_1 = 90\ \mathrm{km}$?
Two particles (1 and 2) move with constant speeds $v_1$ and $v_2$ along two mutually perpendicular straight lines, toward the point where the lines cross. At $t = 0$, particle 1 is a distance $l_1$ from the crossing point and particle 2 is a distance $l_2$ from it. (a) After what time is the distance between the particles smallest? (b) What is that smallest distance?
An airplane must fly due north and cover $s = 300\ \mathrm{km}$ in $t = 1\ \mathrm{h}$. A north-east wind blows with speed $v = 35\ \mathrm{km/h}$ at an angle $\alpha = 40^\circ$ to the meridian. With what speed (relative to the air) must the airplane fly, and what course must it hold?
A motorboat crosses a river of width $d = 1\ \mathrm{km}$. Its speed relative to the water is $v_1 = 4\ \mathrm{km/h}$ and the river flows at $v_2 = 2\ \mathrm{km/h}$. Find: (a) the angle of the boat's path if it is pointed perpendicular to the current; (b) the direction in which the boat must be pointed so that its path is perpendicular to the current; (c) the ratio of the crossing times in the two cases.
Two boats start from the same point with constant velocities $\vec v_1$ and $\vec v_2$ whose directions form an angle $\alpha$. (a) What is the relative velocity of the boats? (b) What is the distance between them a time $t$ after the start?
On the first half of its path, a straight segment at an angle $\alpha_1 = 60^\circ$ to a reference direction, a car moves at $v_1 = 72\ \mathrm{km/h}$. On the second half, a straight segment at $\alpha_2 = 30^\circ$ to the same direction, it moves at $v_2 = 36\ \mathrm{km/h}$. What is the average speed of the car?
During the first half of the travel time, a car moves at $v_1 = 54\ \mathrm{km/h}$ along a straight segment at an angle $\alpha_1 = 45^\circ$ to a reference direction. During the second half of the time, it moves at $v_2 = 72\ \mathrm{km/h}$ along a straight segment at $\alpha_2 = 60^\circ$ to the same direction. What is the average speed of the car?
A car moves at a constant speed $v_0 = 12\ \mathrm{m/s}$. When it brakes, it stops over a distance $s = 60\ \mathrm{m}$. What is the average acceleration of the car while braking?
A body moves with constant acceleration $a = 14\ \mathrm{cm/s^2}$. At time $t_1 = 3\ \mathrm{s}$ after the start of the motion, its speed is $v_1 = 82\ \mathrm{cm/s}$. What was its initial speed, and what distance does it cover in the first $t_2 = 10\ \mathrm{s}$?
A train starts from a station with uniform acceleration. An observer standing beside the track at the front of the train sees the first wagon pass him in $t_1 = 4\ \mathrm{s}$. How long does the 5th wagon take to pass him?
The position vector of a body changes with time as $\vec r(t) = A t^2\,\vec i + B t\,\vec j + C\,\vec k$, where $A = 5\ \mathrm{m/s^2}$, $B = 3\ \mathrm{m/s}$, $C = 2\ \mathrm{m}$. Find: (a) the velocity and acceleration vectors; (b) the speed at $t = 2\ \mathrm{s}$; (c) the magnitude of the acceleration.
The position vector of point A changes with time as $\vec r(t) = bt\,\vec i - ct^2\,\vec j$, where $b$ and $c$ are positive constants. Find: (a) the equation of the trajectory $y = y(x)$; (b) the velocity and acceleration vectors and their magnitudes as functions of time.
The position vector of a body changes with time as $\vec r(t) = (1 - bt)\,kt\,\vec r_0$, where $\vec r_0$ is a unit vector and $b$, $k$ are positive constants. Find: (a) the velocity and acceleration vectors as functions of time; (b) the time $\Delta t$ after which the body returns to its starting position; (c) the distance travelled in that time.
The initial and final velocity vectors of a body are $\vec v_0 = A\vec i + B\vec j + C\vec k$ and $\vec v_h = E\vec i + F\vec j + G\vec k$, with $A = 2$, $B = 5$, $C = 3$, $E = 1$, $F = 5$, $G = 7$ (all in m/s). (a) Draw the velocity vectors in a Cartesian coordinate system. (b) Find the velocity change vector. (c) Find its magnitude.
The equations of motion of a body are $x(t) = bt^2$, $y(t) = ct^2$, with $b = 4\ \mathrm{m/s^2}$ and $c = 2\ \mathrm{m/s^2}$. Find: (a) the equation of the trajectory $y = y(x)$ and sketch it; (b) the speed $v(t)$ and acceleration $a(t)$.
A body moves along the $x$-axis according to $x(t) = A + Bt^2 + Ct^3$, where $A = -1\ \mathrm{m}$, $B = 3\ \mathrm{m/s^2}$, $C = -2\ \mathrm{m/s^3}$. Find: (a) its velocity and acceleration; (b) the time after which it stops; (c) its maximum velocity and when it is reached.
The velocity of a body is $\vec v(t) = A\,\vec i + Bt\,\vec j + Ct^2\,\vec k$, where $A$, $B$, $C$ are constants. Find the change of its position vector between the times $t_1$ and $t_2$.
A body moves in the $xy$-plane with velocity $\vec v = b\,\vec i + cx\,\vec j$, where $b$ and $c$ are positive constants. At the start it was at the origin. Find the equation of the trajectory $y = y(x)$.
A body moves in the positive $x$-direction so that its speed depends on its position as $v(x) = b\sqrt x$, where $b$ is a positive constant. Find: (a) its velocity and acceleration as functions of time $t$; (b) its average speed as a function of the distance travelled $x$.
A body slows down with an acceleration that depends on its speed as $a(v) = -\dfrac{k}{\sqrt v}$, where $k$ is a positive constant. Its initial speed is $v_0$. (a) How long does it take to stop? (b) What distance does it cover in that time?
What is the magnitude of the gravitational force between two electrons that are $r = 0.2\ \mathrm{nm}$ apart? The mass of an electron is $m = 9 \cdot 10^{-31}\ \mathrm{kg}$, and the gravitational constant is $\gamma = 6.67 \cdot 10^{-11}\ \mathrm{N\,m^2/kg^2}$.
The strength of the gravitational field at a certain point is $G = 9.8\ \mathrm{N/kg}$, and the gravitational potential there is $\varphi = -67\ \mathrm{MJ/kg}$. A body of mass $m = 2\ \mathrm{kg}$ is placed at this point. (a) What gravitational force acts on the body? (b) What gravitational potential energy does the body have?
Determine the ratio of (a) the gravitational field strengths and (b) the gravitational potentials at points at a height $h = 37\ \mathrm{km}$ above the Earth's surface and at points on the surface. The Earth's radius is $R = 6374\ \mathrm{km}$.
A thin homogeneous wire of mass $m_1 = 0.1\ \mathrm{kg}$ is bent into a semicircle of radius $r = 0.25\ \mathrm{m}$. A small sphere of mass $m_2 = 0.15\ \mathrm{kg}$ is placed at the centre of the semicircle. Determine the gravitational force between the sphere and the wire.
What minimum energy is needed to transfer an artificial Earth satellite of mass $m = 2\ \mathrm{t}$ from a circular orbit at height $h_1 = R_Z$ to one at height $h_2 = 2R_Z$? The Earth's mean radius is $R_Z = 6370\ \mathrm{km}$ and its mean density $\rho = 5500\ \mathrm{kg/m^3}$.
How much work is needed to move a body of mass $m = 1\ \mathrm{kg}$ from the centre of the Earth to its surface? The Earth's mean radius is $R_Z = 6370\ \mathrm{km}$ and its mean density $\rho = 5500\ \mathrm{kg/m^3}$.
Find how the linear velocity of a point on the Earth's surface depends on the geographic latitude. Calculate this velocity at latitude $\varphi = 45^\circ$. The Earth's mean radius is $R = 6374\ \mathrm{km}$.
(a) What is the linear velocity of the Earth in its orbit around the Sun, if the Sun's mass is $m_S = 2 \cdot 10^{30}\ \mathrm{kg}$ and the Earth–Sun distance is $d = 1.5 \cdot 10^{11}\ \mathrm{m}$? (b) What acceleration does the Sun's gravity give at the Earth?
An aeroplane flies at a height $h = 16\ \mathrm{km}$ with a body of mass $m = 1\ \mathrm{kg}$ on board. The Earth's mean radius is $R = 6374\ \mathrm{km}$ and its mass $M = 6 \cdot 10^{24}\ \mathrm{kg}$. (a) What gravitational force acts on the body? (b) What is the weight of the body if the aeroplane flies at constant velocity over the pole, or over the equator?
Find how the weight of bodies on the Earth's surface depends on the geographic latitude (the effect of the Earth's rotation on the gravitational force).
Neglecting the effect of the Earth's rotation, find how the gravitational force of the Earth on a body depends on the body's distance $r$ from the Earth's centre.
What is the mutual gravitational potential energy of a point mass $m$ and a thin homogeneous rod of mass $M$ and length $l$, lying on the same straight line, if the point is at distance $r_0$ from the near end of the rod?
The period of the Earth's orbit around the Sun is $T = 365$ days and the Earth–Sun distance is $d = 1.5 \cdot 10^{11}\ \mathrm{m}$. Calculate the mass of the Sun, and show that $M_S = 4\pi^2d^3/(\gamma T^2)$.
Estimate the total mechanical energy of the Earth in its elliptical orbit around the Sun. The masses of the Earth and the Sun are about $M_Z = 6 \cdot 10^{24}\ \mathrm{kg}$ and $M_S = 2 \cdot 10^{30}\ \mathrm{kg}$; the greatest Earth–Sun distance is $d = 1.52 \cdot 10^{11}\ \mathrm{m}$.
(a) What velocity must an artificial Earth satellite have to move in a circular orbit at a height $H$ above the Earth's surface? (b) What is its period? (c) What is the first cosmic velocity (the orbital velocity at the Earth's surface), and what is the corresponding period?
Determine the period of the Moon's orbit around the Earth, given the acceleration due to gravity at the Earth's surface $g_0 = 9.83\ \mathrm{m/s^2}$, the Earth's radius $R_Z = 6.37 \cdot 10^6\ \mathrm{m}$, and the Earth–Moon distance $R = 3.8 \cdot 10^8\ \mathrm{m}$.
The Earth's mean density is $\rho = 5500\ \mathrm{kg/m^3}$ and its radius $R = 6.37 \cdot 10^6\ \mathrm{m}$. At what height does an artificial satellite orbit if its period is $T = 90\ \mathrm{min}$ (circular orbit)? Also show that $\left[\sqrt[3]{\gamma\rho T^2/3\pi}\right] = 1$.
The Earth–Moon distance is about $R_{ZM} = 3.85 \cdot 10^8\ \mathrm{m}$ and the Moon's period is $T_M = 27.3$ days. Saturn's moon Dione orbits Saturn at a radius $R_{SD} = 3.78 \cdot 10^5\ \mathrm{m}$ with a period of $T_D = 2.7$ days. From these data, find the ratio of the masses of the Earth and Saturn.
A planet of mass $m$ moves around the Sun in a circular orbit with velocity $v = 34.9\ \mathrm{km/s}$ (relative to the Sun). Find its orbital period, given the Sun's mass. Which planet is it? Also prove the relation $\left[2\pi\gamma M_S/v^2\right] = \mathrm{s}$.
An artificial satellite moves in the Earth's equatorial plane at a distance $R = 2 \cdot 10^7\ \mathrm{m}$ from its centre, from west to east (the same direction as the Earth's rotation). It passes over the same point on the equator every $T_f = 11.6\ \mathrm{h}$. From this, determine the mass of the Earth.
(a) What is the minimum speed a spacecraft must have when leaving the Earth (at a height of 200 km) to reach the Moon? (b) If the spacecraft's mass is $m = 800\ \mathrm{t}$, what minimum kinetic energy (in kWh) is needed at launch?
A body is launched vertically from the Earth's surface with initial velocity $v_0$. Given the Earth's radius $R_Z$ and the surface gravity $g$, find the maximum height the body reaches (no air resistance).
Find the maximum height reached by a rocket launched vertically with an initial velocity equal to the first cosmic velocity. The Earth's radius $R_Z$ and mass $M$ and the gravitational constant are given; neglect air resistance and use energy conservation.
(a) With what velocity does a body falling from outer space reach the Earth, neglecting air resistance? (b) What is its velocity at a height $H = R$ above the Earth's surface, where $R$ is the Earth's radius?
Find the speed at which a point in Belgrade moves because of the Earth's rotation about its axis. The latitude of Belgrade is $\varphi = 44^\circ 45'$, and the radius of the Earth is taken as $R \approx 6470\ \mathrm{km}$.
The minute hand of a clock is $R = 1.2\ \mathrm{m}$ long and the hour hand is $r = 1\ \mathrm{m}$ long. What are the angular velocities of the hands, and what are the speeds of their tips?
A body rotates with frequency $\nu = 100\ \mathrm{Hz}$ along a circle of radius $r = 1\ \mathrm{m}$. (a) What is its speed? (b) What is its normal acceleration? (c) What angular deceleration stops it after three revolutions? (d) What is the magnitude of its total acceleration just after the braking starts?
A wheel of radius $R = 0.5\ \mathrm{m}$ rolls without slipping along a horizontal road. The velocity of its centre is $c = 2\ \mathrm{m/s}$. Find the velocities of the points on the rim that are $\pi/2$ away from the point where the wheel touches the ground.
A car moves along a horizontal circular track of radius $R = 43\ \mathrm{m}$ with a constant tangential acceleration $a = 2\ \mathrm{m/s^2}$. Its initial speed is $v_0 = 36\ \mathrm{km/h}$. How long does the first lap take?
A motor shaft rotates at $\omega_1 = 6000\ \mathrm{rpm}$. During braking, its angular velocity drops uniformly to $\omega_2 = 4800\ \mathrm{rpm}$ in $t = 4\ \mathrm{s}$. Find the angular acceleration of the shaft and the number of revolutions it makes during braking.
The angular velocity of a wheel decreases uniformly from $\omega_1 = 1200\ \mathrm{rpm}$ to $\omega_2 = 600\ \mathrm{rpm}$ in $\Delta t = 10\ \mathrm{s}$. (a) How long does it take for the wheel to stop? (b) How long does its last revolution take?
A body rotating with constant angular velocity $\omega_0 = 4\ \mathrm{rad/s}$ begins to decelerate at $\alpha = 0.5\ \mathrm{rad/s^2}$. Find its angular velocity (a) after $t = 1\ \mathrm{s}$; (b) after turning through $\theta = \pi/3\ \mathrm{rad}$; (c) after $N = 2$ revolutions; and (d) the time after which it stops.
A train moves along a circular track of radius $R = 0.5\ \mathrm{km}$ with angular acceleration $\varepsilon = 0.0049\ \mathrm{rad/s^2}$. What is the train's acceleration at the moment its speed is $v = 60\ \mathrm{km/h}$? What is the angular velocity of its wheels at that moment if their radius is $r = 0.5\ \mathrm{m}$?
A flywheel of radius $R = 0.8\ \mathrm{m}$ rotates with constant angular velocity $\omega_0 = 7.5\ \mathrm{rad/s}$. At some moment the motor driving it stops, and the flywheel keeps turning by inertia for $t = 24\ \mathrm{s}$ before it stops. Find the angular deceleration of the flywheel and the tangential acceleration of a point on its rim during braking.
A car with wheels of diameter $D = 0.60\ \mathrm{m}$ moves along a straight road at $v = 60\ \mathrm{km/h}$. When braking, it stops after $s = 20\ \mathrm{m}$. Assuming uniform deceleration, find the angular deceleration of its wheels during braking.
A disk of radius $R = 12\ \mathrm{cm}$ starts rotating from rest with angular acceleration $\varepsilon = 2\ \mathrm{rad/s^2}$. Find the acceleration of a point on its edge $t = 2\ \mathrm{s}$ after the start.
A wheel of radius $R = 20\ \mathrm{cm}$ starts rotating from rest with constant angular acceleration $\varepsilon = 6.28\ \mathrm{rad/s^2}$. Find the speed and the acceleration of a point on its rim $t = 5\ \mathrm{s}$ after the start.
A material point rotates so that its angular displacement vector changes with time as $\vec\theta(t) = B\,t^{J}\,\vec n_0$, where $B$ is a constant and $\vec n_0$ is the unit vector normal to the plane of the trajectory. Find (a) the angular velocity vector and its magnitude at $t = t_0$; (b) the angular acceleration vector and its magnitude at $t = t_d/6$.
A wheel rotates about a fixed axis so that its angle changes as $\theta(t) = kt^2$, with $k = 1\ \mathrm{rad/s^2}$. Find the total acceleration of a point on its rim at $t = 5\ \mathrm{s}$, if at that moment the point's speed is $v = 1\ \mathrm{m/s}$.
The angle of a rotating body changes with time as $\theta(t) = A + Bt^2$, with $A = 0.1\ \mathrm{rad}$ and $B = 0.2\ \mathrm{rad/s^2}$. (a) What is its angular velocity $t = 3\ \mathrm{s}$ after the start? (b) What is its angular acceleration?
A body moves on a circle with angular velocity that depends on the angle as $\omega(\theta) = \omega_0 - k\theta$, where $\omega_0$ is the initial angular velocity and $k = 2\ \mathrm{s^{-1}}$; $\theta = 0$ at $t = 0$. Find (a) $\theta(t)$; (b) $\omega(t)$; (c) the time after which the angular velocity drops to $\omega_0/2$.
A material point moves along a circle with speed $v(t) = bt$, where $b = 1\ \mathrm{m/s^2}$. What is its total acceleration at the moment when it has turned through $\theta = 0.6\pi\ \mathrm{rad}$?
A body moves along a circle of radius $R$ so that its speed depends on the distance travelled as $v(s) = k\sqrt s$, where $k$ is a constant. Find the angle $\theta$ between the total acceleration and the velocity as a function of $s$.
The normal acceleration of a body moving along a circle of radius $R$ changes with time as $a_n(t) = At + Bt^2$, where $A$ and $B$ are constants. Find (a) the angular velocity as a function of time; (b) the tangential and total acceleration.
The equations of motion of a body are $x = A\cos\omega t$, $y = A\sin\omega t$, $z = Bt$, with $A = 2\ \mathrm{m}$, $B = 4\ \mathrm{m/s}$, $\omega = 5\ \mathrm{rad/s}$. Find the radius of curvature of its trajectory.
The position vector of a particle is $\vec r(t) = (A + Bt^2)\,\vec i + Ct\,\vec j$, where $A$, $B$, $C$ are constants. Find (a) the equation of the trajectory; (b) the velocity and acceleration vectors and their magnitudes; (c) the magnitudes of the tangential and normal components of the acceleration.
A body moves along a circle of radius $R$ so that its tangential acceleration always has the same magnitude as its normal acceleration. At $t = 0$ its speed is $v_0$, and the motion is decelerating. Find (a) the speed as a function of time and of the distance travelled; (b) the total acceleration as a function of speed and of distance.
A smooth metal ring of radius $R = 0.8\ \mathrm{m}$ lies in a horizontal plane. A rod AB rotates about the point A on the ring with constant angular velocity $\omega = 8\ \mathrm{rad/s}$. A small bead C is threaded on both the ring and the rod. Find the velocity and acceleration of the bead.
Two bodies, A and B, fall freely from rest. Body B falls from a height $h_B = 150\ \mathrm{m}$ and lands $\Delta t = 3.5\ \mathrm{s}$ earlier than body A. From what height was body A released?
A helicopter hovers at a height $H = 980\ \mathrm{m}$. A bomb is released from it and falls freely. After what time does the pilot hear the explosion? The speed of sound is $c = 340\ \mathrm{m/s}$.
A stone is dropped into a chasm and falls freely. The sound of it hitting the water is heard $t = 4.9\ \mathrm{s}$ after it was released. How deep is the chasm? The speed of sound is $c = 340\ \mathrm{m/s}$.
A steel ball is dropped from a height $h = 1\ \mathrm{m}$ onto a steel plate and bounces. At each bounce its speed is reduced by 10 %. How long does its third fall last?
A body falls freely from rest. At point A its speed is $v_A = 29.43\ \mathrm{m/s}$, and at a lower point B it is $v_B = 49.05\ \mathrm{m/s}$. Find the height difference between A and B and the time the body needs to get from A to B.
Two bodies are released at the same time from a height $H = 7900\ \mathrm{m}$. One starts from rest; the other is given an initial downward speed $v_0 = 200\ \mathrm{m/s}$. How does the distance between them change with time? What is it when the second body reaches the ground? (Air resistance is ignored.)
One body is released from rest at a height $H = 8000\ \mathrm{m}$. At the same moment another body is thrown vertically upward from the ground, along the same line, with initial speed $v_0$. What must $v_0$ be for the bodies to meet halfway, at $H/2$?
One body is released from rest at a height $H = 8000\ \mathrm{m}$. At the same moment another body is thrown vertically upward from the ground along the same line with speed $v_0$. What must $v_0$ be for the bodies to meet at $H/2$? What is the velocity of the first body relative to the second when they meet?
A body falls freely from rest and covers the second half of its total height in $\Delta t = 1\ \mathrm{s}$. (a) What is its total fall time? (b) From what height was it released?
A body is thrown vertically upward with initial speed $v_0 = 10\ \mathrm{m/s}$. (a) How high does it rise? (b) After what time does it fall back to the ground? (c) With what velocity does it hit the ground?
A body is thrown vertically upward with speed $v_0 = 10\ \mathrm{m/s}$ from the top of a tower of height $H = 25\ \mathrm{m}$. (a) How long does it take to hit the ground? (b) With what velocity does it hit the ground?
Two spheres start falling from the same height at the same moment. One starts from rest; the other has an initial downward speed $v_0 = 20\ \mathrm{m/s}$. The first sphere lands $\Delta t = 2\ \mathrm{s}$ after the second. (a) From what height were they released? (b) How long did each fall?
A helicopter hovers $H = 400\ \mathrm{m}$ above a lake. A bomb is released and falls through the air (air resistance neglected). In the water it keeps moving downward with a constant deceleration of $a = 4.5\ \mathrm{m/s^2}$. It explodes on reaching the bottom, $t = 22\ \mathrm{s}$ after release. How deep is the lake?
A body falls freely from rest. In the last $\Delta t = 0.6\ \mathrm{s}$ of its fall it covers $\Delta s = 30\ \mathrm{m}$. From what height was it dropped?
A stone is thrown down into a well with initial speed $v_0 = 20\ \mathrm{m/s}$. The sound of it hitting the water is heard $t = 2\ \mathrm{s}$ after it was thrown. The speed of sound is $c = 340\ \mathrm{m/s}$. How deep is the well?
A balloon rises vertically at $v_0 = 5\ \mathrm{m/s}$. At a height of $H = 30\ \mathrm{m}$ a load is released from it with zero velocity relative to the balloon. (a) At what height is the load $t_1 = 2\ \mathrm{s}$ after release? (b) How long does it take to reach the ground?
Two objects are thrown at the same moment from the same point on the ground with the same speed $v_0 = 30\ \mathrm{m/s}$: one vertically upward, the other at $\theta = 45^\circ$ to the horizontal. What is the distance between them $t = 2\ \mathrm{s}$ later?
A pilot's body can withstand a maximum acceleration of $a_{\max} = 4.6\,g$. What is the smallest radius of curvature of the plane's path when pulling out of a dive, if the plane flies at a constant speed of $v = 200\ \mathrm{km/h}$?
What horizontal speed must be given to a body at the equator, in the direction of the Earth's rotation, so that it moves parallel to the Earth's surface (i.e. keeps its height)? At the equator $g = 9.78\ \mathrm{m/s^2}$ and the Earth's radius is $R = 6378.4\ \mathrm{km}$.
A cannon on a hill $H = 120\ \mathrm{m}$ above a lake fires a projectile horizontally toward the lake with speed $v_0 = 320\ \mathrm{m/s}$. Find where the projectile hits the water, its speed on impact, and the angle at which it strikes the surface.
An airplane flies horizontally at a constant speed $v = 250\ \mathrm{km/h}$ and releases a bomb. Find (a) the position of the bomb $t = 10\ \mathrm{s}$ later, relative to the release point; (b) its speed at that moment; (c) the direction of its motion relative to the horizontal at that moment.
From three points on a vertical shore, three identical balls are launched horizontally at the same moment with speeds $v_{01} = 50$, $v_{02} = 75$ and $v_{03} = 100\ \mathrm{m/s}$. The first ball hits the water at a horizontal distance $D_1 = 100\ \mathrm{m}$. All three hit the water at the same time. Find (a) the flight times; (b) the launch heights; (c) the speeds on impact.
An airplane flies at a constant altitude $H = 2000\ \mathrm{m}$ with horizontal speed $v_0 = 300\ \mathrm{m/s}$. Above point A it releases a bomb. Find (a) the time until the bomb hits the ground; (b) the horizontal distance AB it covers before landing.
An airplane and a ship move in the same vertical plane along parallel straight lines: the ship with speed $v_1$, the airplane with speed $v_2$, at a height $h$ above the ship. At what horizontal distance from the ship must the bomb be released to hit it, if (a) they move in the same direction; (b) they move in opposite directions?
A body is thrown horizontally from a height $h = 6\ \mathrm{m}$ above level ground and lands $l = 10\ \mathrm{m}$ away horizontally. Find (a) its initial speed; (b) the angle at which it hits the ground; (c) the equation of its trajectory.
A body is thrown at an angle $\alpha = 30^\circ$ to the horizontal. After $t = 2\ \mathrm{s}$ its speed is $v = 400\ \mathrm{m/s}$. Find the initial speed $v_0$.
A body is thrown with speed $v_0 = 200\ \mathrm{m/s}$ at $\alpha = 60^\circ$ to the horizontal. Find the velocity components $v_x$ and $v_y$: at the start, at the highest point, and when it hits the ground.
A projectile is fired from a cannon with speed $v_0 = 400\ \mathrm{m/s}$ at $\alpha = 30^\circ$ to the horizontal. Find its horizontal range and its time of flight (level ground, no air resistance).
Two projectiles are fired with the same speed, one at $\alpha_1 = 45^\circ$ and the other at $\alpha_2 = 30^\circ$ to the horizontal. Find the ratios of (a) their maximum heights; (b) their ranges; (c) their times of flight.
A rocket is launched at $\alpha = 70^\circ$ to the horizontal and reaches its highest point after $t_m = 80\ \mathrm{s}$. Find its initial speed and the horizontal distance to where it lands (assume free flight after launch, level ground, no air resistance).
A body is thrown with speed $v_0$ at an angle $\alpha$ to the horizontal. Find the angle its velocity makes with the horizontal at a horizontal distance $x$ from the launch point.
At what angle must a body be thrown so that (a) its maximum height equals its horizontal range; (b) its horizontal range equals the height it would reach if thrown vertically upward with the same speed?
At what height does a shell hit a vertical wall $x = 4\ \mathrm{km}$ from the cannon? The shell is fired at $v_0 = 400\ \mathrm{m/s}$ and $\alpha = 14^\circ$ to the horizontal (no air resistance).
A shell is fired at $v_0 = 200\ \mathrm{m/s}$ and $\alpha = 45^\circ$ to the horizontal. For what time must its fuse be set so that it explodes at a height of $h = 10\ \mathrm{m}$ on the way down, before hitting the ground?
A rocket of mass $m = 8400\ \mathrm{kg}$ is launched from the ground at $v_0 = 350\ \mathrm{m/s}$ and $\alpha = 80^\circ$ to the horizontal. At the highest point of its path a part of mass $m_1 = 40\ \mathrm{kg}$ is ejected backward with speed $v = 200\ \mathrm{m/s}$ relative to the rocket. What is the rocket's speed immediately after the ejection?
A ship moves in a straight line with constant speed $v$. A projectile is launched from the ship with speed $v_0$ (relative to the water) in the direction of the ship's motion, at an angle $\alpha$ to the horizontal. (a) How far from the ship does the projectile land in the water? (b) At what angle $\alpha_2$ must it be launched to land back on the ship? Give the trajectory in that case. (c) What is the flight time in each case? (d) In case (b), what must the ship's speed be for the projectile's maximum height to equal the distance the ship travels until the projectile lands on it? Ignore the ship's size.
A mortar fires shells at a target B with initial speed $v_0 = 150\ \mathrm{m/s}$. The target is hit both with the launch angle $\alpha = 30^\circ$ and with $\beta = 60^\circ$. How far away is the target (level ground, no air resistance)?
A tank drives straight toward a cannon at a constant $v = 3.6\ \mathrm{km/h} = 1\ \mathrm{m/s}$. With what initial speed must a shell be fired at $\alpha = 30^\circ$ to hit the tank, if the tank was $d = 8\ \mathrm{km}$ away at the moment of firing?
From a seaside cliff of height $H = 80\ \mathrm{m}$, a body is launched at $v_0 = 400\ \mathrm{m/s}$ and $\alpha = 45^\circ$ to the horizontal. Where does it hit the water, and how long is it in the air (no air resistance)?
A rod of length $l = 1\ \mathrm{m}$ rotates with constant angular velocity $\omega = 314\ \mathrm{rad/s}$ about an axis through one end. A small object comes off the other end and flies off along the tangent, at $\alpha = 45^\circ$ to the horizontal. (a) How far away does it land on the horizontal plane? (b) How high does it rise? (Treat the launch point as being on the plane; no air resistance.)
A rocket starts vertically upward with constant acceleration $a_1 = 4g$. After $t_0 = 8\ \mathrm{s}$ a part of it separates and is pushed sideways with speed $v_0 = 80\ \mathrm{m/s}$. Where does this part land? Ignore air resistance.
A bomb is released from an airplane flying at a height $H = 1\ \mathrm{km}$ with speed $v_0 = 720\ \mathrm{km/h}$. To what time must its timer be set so that it explodes $h = 100\ \mathrm{m}$ above the ground?
From a cliff of height $H = 5\ \mathrm{m}$, a stone is thrown horizontally with $v_0 = 25\ \mathrm{m/s}$. When it hits the water it bounces off at the same angle at which it arrived (without losing speed). Where does it hit the water the second time?
An object is thrown from a seaside cliff with speed $v_0 = 10\ \mathrm{m/s}$ at $\alpha = -30^\circ$ (30° below the horizontal). Find its position and velocity $t = 2\ \mathrm{s}$ later.
What is the smallest speed with which an object must be thrown to land $D = 64\ \mathrm{m}$ away? Launch and landing are at the same height; air resistance is neglected.
A rifle bullet passes through two vertical sheets of paper $d = 20\ \mathrm{m}$ apart. The hole in the second sheet is $h = 2\ \mathrm{cm}$ lower than the hole in the first. The bullet was moving horizontally at the first sheet. Find its speed.
From point A at the foot of an inclined plane with angle $\theta = 30^\circ$, an object is thrown up the slope with speed $v_0 = 5\ \mathrm{m/s}$ at $\alpha = 45^\circ$ to the horizontal. (a) At what distance $l$ from A (along the slope) does it land on the inclined plane? (b) What is the greatest height it reaches above the horizontal?
Two vertical blocks of height $h = 0.5\ \mathrm{m}$ leave a narrow vertical gap of width $d = 3\ \mathrm{cm}$. A small steel ball of radius $R = 0.3\ \mathrm{cm}$ rolls toward the gap, perpendicular to it, with speed $v_0 = 1\ \mathrm{m/s}$, falls in, and bounces elastically between the walls until it reaches the bottom. How many times does it hit the walls?
What is the smallest speed with which a stone must be thrown, from a height $h$, to get over a rectangular obstacle of width $l$ and height $H$ ($H \gt h$)? The thrower may choose both the angle and the distance from the obstacle.
A body falls from a height $h$ onto an inclined plane with angle of inclination $\alpha$. Each time it hits the plane it bounces off elastically (the speed keeps its magnitude and the angle of reflection equals the angle of incidence). Find the distances $x_1, x_2, \dots, x_n$ between successive points where it hits the plane.
An airplane flies horizontally at a constant speed $v = 250\ \mathrm{km/h}$ and releases a bomb. Find, $t = 10\ \mathrm{s}$ after the release, (a) the position of the bomb, (b) its speed, and (c) the direction of its motion relative to the horizontal. Ignore air resistance and assume the bomb has not yet reached the ground.
A quartz nugget containing gold has a mass $m = 51\ \mathrm{g}$ and an average density $\rho = 8500\ \mathrm{kg/m^3}$. How much gold does it contain? (Densities: quartz $\rho_1 = 2650\ \mathrm{kg/m^3}$, gold $\rho_2 = 19\,300\ \mathrm{kg/m^3}$.)
In what ratio must distilled water and alcohol be mixed to obtain a liquid of density $\rho = 800\ \mathrm{kg/m^3}$? When water and alcohol are mixed, the total volume decreases by 3% compared with the sum of the volumes of the components. (Densities: water $\rho_w = 1000$, alcohol $\rho_a = 789\ \mathrm{kg/m^3}$.)
A cube with edge $a = 10\ \mathrm{cm}$ is made of a metal of density $\rho = 8600\ \mathrm{kg/m^3}$. Find (a) its mass on Earth and on the Moon, and (b) its weight on the surface of Earth and of the Moon. ($g_E = 9.81\ \mathrm{m/s^2}$, $g_M = 1.62\ \mathrm{m/s^2}$.)
A body of mass $m = 10\ \mathrm{kg}$ is taken from Earth to the Moon. Find: (a) the gravitational force on it on Earth and on the Moon; (b) the reading of a dynamometer at rest from which it hangs, on Earth and on the Moon; (c) its weight when it lies on the ground of each body; (d) its weight in a rocket launched from Earth, at the moment the rocket accelerates vertically upward with $a = 4g$; (e) its weight in a rocket launched from the Moon with the same acceleration. ($g_E = 9.81\ \mathrm{m/s^2}$, $g_M = 1.62\ \mathrm{m/s^2}$.)
Two balls of equal mass $m_1 = m_2 = 2\ \mathrm{g}$ move with equal speeds $v_1 = v_2 = 20\ \mathrm{cm/s}$ along mutually perpendicular paths. What is the total momentum of the two balls?
A ball of mass $m = 20\ \mathrm{g}$ falls from a height $h = 2\ \mathrm{m}$ onto a plane inclined at $\alpha = 45^\circ$. It bounces off horizontally, without any change in the magnitude of its velocity. What is the change in the ball's momentum?
A force $F = 20\ \mathrm{N}$ acts on a body of mass $m = 200\ \mathrm{g}$ for $\Delta t = 0.03\ \mathrm{s}$. What is the impulse of the force, and what is the body's speed after the force stops, if it started from rest?
A force $F = 4\ \mathrm{N}$ acts on a body for $\Delta t = 4\ \mathrm{s}$, and the body's speed changes by $\Delta v = 2\ \mathrm{m/s}$. What is its mass?
A ball of mass $m = 20\ \mathrm{g}$ hits a wall perpendicularly at $v = 10\ \mathrm{cm/s}$ and bounces straight back with the same speed. The collision lasts $\Delta t = 0.1\ \mathrm{s}$. Find the impulse the wall gives the ball and the magnitude of the average force of the wall on the ball.
A metal ball of mass $m = 10\ \mathrm{g}$ falls freely from a height $H = 30\ \mathrm{m}$ onto a smooth metal plate and bounces back with the same speed. The contact lasts $\Delta t = 1\ \mathrm{ms}$. Find the magnitude of the impulse the plate gives the ball and the average force of the ball on the plate.
Two boats float on a lake: the first has mass $m_1 = 200\ \mathrm{kg}$, the second $m_2 = 450\ \mathrm{kg}$. A rope is stretched between them, and a person in the first boat pulls it with a force $F = 200\ \mathrm{N}$. Find the speed of the first boat relative to the shore and relative to the second boat $t = 2\ \mathrm{s}$ after the pulling starts. The boats start at rest; neglect water resistance.
A cannon of mass $m_1 = 1.8\ \mathrm{t}$ fires a shell of mass $m_2 = 1\ \mathrm{kg}$ at $\alpha = 30^\circ$ to the horizontal with speed $v_2 = 360\ \mathrm{m/s}$. Find (a) the recoil speed of the cannon and (b) its average deceleration if it stops after sliding $s = 2\ \mathrm{m}$.
A railway wagon of mass $m_1 = 8\ \mathrm{t}$, standing still, carries a rocket launcher. Two rockets of mass $m_2 = 80\ \mathrm{kg}$ each are launched at the same time with speed $v_0 = 1000\ \mathrm{m/s}$, horizontally along the track. How far does the wagon move because of the launch, if the total friction coefficient during its motion is $\mu = 0.06$?
A shell moving horizontally at $v_0 = 15\ \mathrm{m/s}$ explodes into two fragments of masses $m_1 = 0.5\ \mathrm{kg}$ and $m_2 = 1\ \mathrm{kg}$. The larger fragment moves at $v_2 = 30\ \mathrm{m/s}$ along the original direction of the shell. Find the velocity of the smaller fragment, which moves along the same line.
A helicopter of mass $m_1 = 2.5\ \mathrm{t}$ hovers in the air. A man of mass $m_2 = 80\ \mathrm{kg}$ climbs a rope ladder hanging from it at a constant speed $v = 0.5\ \mathrm{m/s}$ relative to the ladder. With what velocity, and in which direction, does the helicopter move?
A rope is passed over a pulley of radius $R$. Two people of equal mass $m_A = m_B = m$ hang from its ends, A on one side and B on the other, at rest. What happens to B if A starts climbing the rope at a constant speed $v$ relative to the rope? Neglect the masses of the rope and pulley, and friction.
Two identical boats, each of mass $m = 180\ \mathrm{kg}$ (including one package), move on parallel courses toward each other, each at $v_0 = 3\ \mathrm{m/s}$ relative to the shore. (1) At one meeting, a package of mass $m_1 = 20\ \mathrm{kg}$ is passed from the first boat to the second, and then the same package is passed back. (2) At another meeting, the boats exchange identical packages simultaneously. Water resistance is neglected, and each package is handed across sideways (it keeps the velocity of the boat it leaves). After which exchange are the boats' speeds greater?
A grenade moving at $v = 400\ \mathrm{m/s}$ breaks into two equal parts. Each part moves along a straight line at $30^\circ$ to the original direction of motion (one on each side). Find the speeds of the parts.
At the launch of a rocket of mass $m = 200\ \mathrm{kg}$, a quarter of its mass burns instantly and is ejected backward as gas, at $v_1 = 1800\ \mathrm{m/s}$ relative to the rocket. Find (1) the rocket's speed after the ejection, and (2) how far it flies if it is launched at $\alpha = 30^\circ$ to the horizontal (no air resistance, level ground).
A body of mass $m = 200\ \mathrm{g}$ on a rope of length $l = 0.5\ \mathrm{m}$ rotates in a vertical plane. The rope breaks when its tension reaches $F_{\max} = 295\ \mathrm{N}$. What is the greatest angular velocity at which the body can rotate without the rope breaking?
A car of mass $m$ moves at constant speed $v$ (a) on a horizontal road, (b) over a convex bridge with radius of curvature $R$, (c) through a concave bridge with the same radius. Find the force the car exerts on the road (its weight) in each case, when the car is at the middle of the bridge.
A body of mass $m = 1\ \mathrm{kg}$ on a rope of length $l = 0.5\ \mathrm{m}$ rotates in a vertical plane with constant angular velocity $\omega = 10\ \mathrm{rad/s}$. Find the rope tension at the lowest point A, the highest point B, and the two points C and D where the rope is horizontal.
An airplane flying at $v = 600\ \mathrm{km/h}$ makes a vertical loop of radius $R = 600\ \mathrm{m}$. With what force does the pilot, of mass $m = 80\ \mathrm{kg}$, press on the seat at the highest and lowest points?
Two identical bodies of mass $m = 0.5\ \mathrm{kg}$ are fixed on opposite sides of a motor shaft rotating at $n = 6000\ \mathrm{rpm}$, each at a distance $R = 0.3\ \mathrm{m}$ from the axis, so the shaft is balanced. Then one body moves 10% farther out. Find the force this unbalanced system exerts on the shaft. Neglect the mass of the rod.
Two bodies of equal mass $m$ are connected by a rope passing through a hole in a horizontal table. One body moves on the table in a circle of radius $R$ around the hole; the other hangs below. What angular velocity must the circling body have for the hanging body to stay at the same height? Neglect all friction.
A simple pendulum of length $l$ is pulled aside by $\theta = 60^\circ$ and released. As it passes through the equilibrium position, its string catches on a peg O located $l/2$ below the point of suspension. By how much does the string tension change when the string hits the peg?
A little sand lies at the bottom of a hollow sphere of inner radius $R$. Where does the sand collect when the sphere rotates about a vertical axis with angular velocity $\omega$? Neglect friction between the sand and the wall.
To measure the coefficient of friction between asphalt and rubber, a rubber-covered body of mass $m = 100\ \mathrm{kg}$ is placed on the road. A horizontal pulling force $F = 196\ \mathrm{N}$ moves it uniformly along the road. What is the coefficient of friction?
A man of mass $m = 85\ \mathrm{kg}$ stands on the floor of a tram moving with constant acceleration $a = 3\ \mathrm{m/s^2}$. The coefficient of friction between the floor and his shoes is $\mu = 0.1$. Find: (a) the inertial force on the man; (b) the force he must supply by holding a handle so as not to slide; (c) how hard he would have to press down on the floor (without a handle) so as not to slide.
A body of mass $m_2 = 4\ \mathrm{kg}$ lies on a wheeled platform of mass $m_1 = 40\ \mathrm{kg}$; the coefficient of friction between them is $\mu = 0.2$. A horizontal force acts on the body. With $F_1 = 2\ \mathrm{N}$ the body does not move relative to the platform; with $F_2 = 100\ \mathrm{N}$ it slides. Find the friction force and the platform's acceleration in both cases. The platform rolls without friction.
A small ball of mass $m = 20\ \mathrm{g}$ is held against the front of a vertical plate that moves horizontally with constant acceleration $a_1 = 4g$. (a) What force does the ball exert on the plate? (b) If the coefficient of friction between the ball and the plate is $\mu = 0.5$, at what acceleration $a_2$ does the ball start to slide down? (c) If the plate's acceleration is $a_3 = g$, how long does it take the ball to slide down a height $h = 10\ \mathrm{cm}$?
A tractor of mass $m_2$ drags a rock of mass $m_1 = 7\ \mathrm{t}$ over level ground. The coefficient of friction between the rock and the ground is $\mu_1 = 0.4$; between the tractor's wheels and the ground it is $\mu_2 = 0.2$. What is the smallest tractor mass for which it can drag the rock?
How far can a cyclist lean (tilt the bicycle from the vertical) when riding through a curve without the risk of falling? The coefficient of friction between the tyres and the road is $\mu$.
A motorcyclist rides on the inside wall of a vertical cylinder ("wall of death") of radius $R = 15\ \mathrm{m}$. The coefficient of friction between the tyres and the wall is $\mu = 0.4$. What is the smallest speed at which he does not slide down?
What is the highest speed at which a car can drive around a horizontal circular path of radius $R = 30\ \mathrm{m}$ without skidding, if the coefficient of friction between the tyres and the road is $\mu = 0.25$?
A body of mass $m = 20\ \mathrm{kg}$ lies on a horizontal platform rotating with constant angular velocity $\omega = 10\ \mathrm{rad/s}$, at a distance $R = 0.5\ \mathrm{m}$ from the axis. (a) Find the centrifugal force on the body. (b) At what angular velocity does the body start to slide, if the coefficient of friction is $\mu = 0.2$?
A crate of mass $m = 1200\ \mathrm{kg}$ stands on the open platform of a truck. What is the largest acceleration the truck can have without the crate sliding on the platform? The coefficient of friction is $\mu = 0.3$.
A metal plug of length $l = 4\ \mathrm{cm}$ was inserted into a long metal pipe of radius $r = 2\ \mathrm{cm}$ while the pipe was hot. After cooling, the pipe presses on the plug with a pressure $p = 40\ \mathrm{kPa}$. What axial force is needed to pull the plug out, if the coefficient of friction is $\mu = 0.3$? Assume no permanent deformation.
A train (locomotive and cars) has a total mass $m = 400\ \mathrm{t}$; the coefficient of friction (rolling resistance) is $\mu = 0.04$. What tractive force must the locomotive exert for the train to reach $v = 40\ \mathrm{km/h}$ from rest in $t = 120\ \mathrm{s}$?
To find the coefficient of friction between two pieces of oak, a wooden cube of mass $m = 2\ \mathrm{kg}$ is placed on an oak board, and the board is slowly tilted while being tapped lightly with a hammer. At an angle of $\alpha = 32^\circ$ the cube starts to slide down uniformly. What is the coefficient of friction?
A car climbs a slope of angle $\alpha = 10^\circ$ at a steady speed $v_1$. Coming down the same slope with the same engine power, its steady speed is $v_2 = 30\ \mathrm{m/s}$. What will its speed be on a level road under the same conditions (same power, same resistance)?
A small body slides down a helical (spiral) groove of radius $R$ and pitch $h$ (the drop per turn). The coefficient of friction between the body and the groove is $\mu$. With what steady speed does the body eventually slide down?
A train of mass $m = 400\ \mathrm{t}$ moves on rails; the friction force is $6/1000$ of its weight. What speed does the train reach if a force $F = 60\ \mathrm{kN}$ acts on it for $\Delta t = 50\ \mathrm{s}$, starting from rest?
A stone block of mass $m = 200\ \mathrm{kg}$ lies on an incline of angle $\alpha = 15^\circ$. To move it down the incline at constant speed, a force $F = 490\ \mathrm{N}$ along the incline is needed. (a) What is the coefficient of friction? (b) What force along the incline moves the block up at constant speed?
A constant vertical force $F = 10.81\ \mathrm{N}$ directed upward acts on a body of mass $m = 1\ \mathrm{kg}$, initially at rest, for $t = 10\ \mathrm{s}$. To what height does the body rise?
A body of mass $m = 160\ \mathrm{g}$ is thrown vertically upward at $v_0 = 100\ \mathrm{m/s}$ and reaches its highest point after $t = 80\ \mathrm{s}$. What is the average air resistance force?
Under the action of a constant force $F$, a body of mass $m = 50\ \mathrm{g}$, starting from rest, travels $s = 30\ \mathrm{m}$ in $t = 2\ \mathrm{s}$. How large is the force?
A mine elevator (cage) of mass $m = 600\ \mathrm{kg}$ may move with accelerations up to $a = \pm 1.2\ \mathrm{m/s^2}$. Find (a) the greatest tension in the cable; (b) the tension if $a = g$; (c) the tension when the cage is at rest and when it moves at constant speed.
A body of mass $m = 50\ \mathrm{kg}$ moves at $v_0 = 4\ \mathrm{m/s}$. A constant force $F = 30\ \mathrm{N}$ acts on it for $t = 2\ \mathrm{s}$, opposite to its motion. (a) By how much does its speed decrease? (b) How large would the force have to be to stop the body in the same time?
A force whose time dependence is shown in Figure III acts on a body of mass $m = 100\ \mathrm{kg}$. Draw the velocity and acceleration diagrams of the body if (a) it was initially at rest, (b) it initially moved at $v = 10\ \mathrm{m/s}$ opposite to the force.
A constant force $F = 100\ \mathrm{N}$ acts on a body of mass $m = 5\ \mathrm{kg}$, initially at rest, for the time shown in Figure III. (a) What is the acceleration while the force acts? (b) What speed does the body reach? (c) Draw the velocity and acceleration diagrams.
What net force must act on an elevator of mass $m = 1000\ \mathrm{kg}$ to give it an acceleration $a = 2\ \mathrm{m/s^2}$? What cable forces are needed to accelerate it upward and downward with this acceleration?
A balloon with total mass $m$ (including its load) falls with acceleration $a$ under the action of the lift (buoyant) force $F_p$. What mass $\Delta m$ of ballast must be dropped for the balloon to rise with an acceleration of the same magnitude? (Neglect air resistance; the lift stays the same.)
Two bodies of masses $m_1 = 1\ \mathrm{kg}$ and $m_2 = 5\ \mathrm{kg}$ are connected by a light inextensible rope as shown in the diagram; one end of the rope is fixed. The body $m_1$ slides down a frictionless incline of angle $\alpha = 30^\circ$. Find the acceleration $a_2$ of the body $m_2$. Neglect friction and the mass of the pulley.
A body of mass $m$ moves according to $x = A\sin\omega t$, $y = B\cos\omega t$, $z = 0$, where $A$, $B$ and $\omega$ are constants. Find its velocity, acceleration and the force acting on it.
A force $F = A\sin\omega t$ ($A$, $\omega$ constants) starts acting on a body of mass $m$ at rest. (a) Find the velocity as a function of time. (b) Find the distance travelled in time $t$.
A force $F = A\sin\omega t$ ($A$, $\omega$ constants) starts acting on a body of mass $m$ at rest. (a) Find the velocity as a function of time. (b) Find the distance travelled in time $t$.
A body of mass $m = 1\ \mathrm{kg}$, at rest on a smooth horizontal surface, is acted on by a force of constant magnitude $F = mg/2$. As the body moves in a straight line, the angle $\theta$ between the force and the horizontal changes as $\theta = \pi s/k$, where $s$ is the distance travelled and $k = 8\ \mathrm{m}$. What is the body's speed when $\theta = \pi/2$?
A force $F = kt^2$ with $k = 10\ \mathrm{N/s^2}$ acts on a body of mass $m = 1\ \mathrm{kg}$, initially at rest, for $\tau = 3\ \mathrm{s}$. (a) What is the body's momentum when the force stops? (b) How far does it travel while the force acts?
A small body of mass $m$ lies on a smooth horizontal surface. From $t = 0$ a force $\vec F$ acts on it; its magnitude grows as $F = kt$ ($k$ constant) and its direction is fixed, at an angle $\theta$ above the horizontal. (a) What is the body's speed when it leaves the surface? (b) How far has it travelled by then?
A boat of mass $m_1$ moves on a lake at speed $v_0$ when its engine is switched off. The water resistance is proportional to the speed, $F = -kv$. Find (1) how long the boat keeps moving, (2) its speed as a function of the distance travelled, (3) the total distance it covers.
A bullet of mass $m$ passing through a wooden plank of thickness $D = 5\ \mathrm{cm}$ slows from $v_0 = 350\ \mathrm{m/s}$ to $v_1 = 250\ \mathrm{m/s}$. The resistance force is proportional to the square of the speed, $F = -kv^2$. Find (1) the bullet's speed as a function of the distance travelled in the plank, (2) the time it takes to pass through.
A train of $n$ identical wagons, each of mass $m$, is pulled by a force $F$. Neglecting friction, find the tension in the couplings between the wagons.
Two bodies of masses $m_1 = 10\ \mathrm{kg}$ and $m_2 = 20\ \mathrm{kg}$, connected by a rope, lie on a horizontal surface. The body $m_2$ is pulled with a force $F = 20\ \mathrm{N}$. Find the acceleration and the rope tension. Neglect friction.
Two bodies of masses $m_1 = 2\ \mathrm{kg}$ and $m_2 = 3\ \mathrm{kg}$, on a frictionless horizontal surface, are connected by a rope that breaks at a tension of $T_{\max} = 350\ \mathrm{N}$. Forces growing with time, $F_1 = kt$ on $m_1$ and $F_2 = 2kt$ on $m_2$, with $k = 10\ \mathrm{N/s}$, pull the bodies in opposite directions (apart). When does the rope break?
In the system shown in diagram E, the masses are $m_1 = 10\ \mathrm{kg}$ and $m_2 = 5\ \mathrm{kg}$; the coefficient of friction between $m_1$ and its surface is $\mu = 0.2$, and the incline angle is $\alpha = 30^\circ$. Find (a) the acceleration of the system and (b) the rope tension.
A body slides from rest down an incline of length $l = 30\ \mathrm{m}$ and angle $\theta = 30^\circ$. The coefficient of friction is $\mu = 0.17$. (a) How long does it take to slide the whole length? (b) What is its speed at the bottom?
Two bodies of masses $m_1 = 200\ \mathrm{g}$ and $m_2 = 300\ \mathrm{g}$ hang from the ends of a rope passing over a frictionless pulley of negligible mass. Find the acceleration of the system.
Two bodies of equal mass $m$ hang on a rope over a pulley. An extra mass $\Delta m$ is placed on one of them. Find (1) the acceleration, (2) the rope tension, (3) the force with which the added mass presses on the body it lies on, (4) the force on the pulley's axle. The pulley is massless and frictionless.
A body of mass $m = 500\ \mathrm{kg}$ on an incline of angle $\alpha = 30^\circ$ (friction coefficient $\mu = 0.1$) is moving down the incline at $v_0 = 2\ \mathrm{m/s}$. What force $F$ along the incline (pointing up the slope) will stop it in $t_0 = 5\ \mathrm{s}$?
A body of mass $m = 2\ \mathrm{kg}$ lies on an incline of angle $\alpha = 30^\circ$ with friction coefficient $\mu = 0.2$. What is the smallest horizontal force that keeps it from sliding down?
Bodies of masses $m_1 = 3\ \mathrm{kg}$ and $m_2 = 2\ \mathrm{kg}$ lie on a smooth incline of angle $\alpha = 30^\circ$, joined by a spring of stiffness $k = 1000\ \mathrm{N/m}$. A rope from $m_2$ passes over a pulley at the top to a hanging body $m_3 = 9\ \mathrm{kg}$. Find the extension of the spring while the system moves.
Three bodies of masses $m_1 = 4\ \mathrm{kg}$, $m_2 = 16\ \mathrm{kg}$ and $m_3 = 8\ \mathrm{kg}$ are connected by ropes and arranged as shown in the figure; the coefficient of friction with the surface is $\mu = 0.1$. Find the direction of motion and the acceleration.
A body of mass $m_1 = 1\ \mathrm{kg}$ is pulled upward by a body of mass $m_2 = 9\ \mathrm{kg}$ sliding down an incline of angle $\alpha = 60^\circ$; the rope runs over a pulley at the top. The coefficient of friction between $m_2$ and the incline is $\mu = 0.2$. Neglecting the masses of the pulley and rope, find (a) the acceleration of $m_2$, (b) the support forces (at A and B in the figure).
A body of mass $m_1 = 2\ \mathrm{kg}$ lies on a body of mass $m_2 = 1\ \mathrm{kg}$, which rests on a smooth horizontal table. The two are connected by a light rope passing around a pulley fixed to a wall. What force must act on the lower body so that it moves with acceleration $a = g/2$? The coefficient of friction between the bodies is $\mu = 0.5$; the table is frictionless.
A small body starts sliding from rest down an incline of angle $\alpha$. The coefficient of friction grows with the distance travelled as $\mu = ks$ ($k$ constant). (a) How far does the body go before it stops? (b) What is its greatest speed?
A small body of mass $m_1$ slides down a rough inclined plane of angle $\alpha$ and length $l$, which is fixed on a cart; the cart with the incline has mass $m_2$. The coefficient of friction between the body and the incline is $\mu$ (small enough for the body to slide); the cart rolls without friction. The system starts from rest. How far does the cart move while the body slides from the top of the incline to the bottom?
A jet aircraft of mass $m$ hovers at a fixed height by blowing its exhaust gases straight down. The exhaust opening has area $S$, and the gas density is $\rho$. At what speed must the gases leave the engine?
A container with water, of total mass $m$, stands on a horizontal surface and moves at constant velocity because water flows out through a small opening in its side. The water level is $h$ above the opening, and the opening has area $S$. What is the coefficient of friction between the container and the surface? Assume $h$ stays constant.
A rocket with initial mass $m_0 = 600\ \mathrm{t}$ burns fuel at a rate $k = 3\ \mathrm{t/s}$; the exhaust speed relative to the rocket is $u = 2000\ \mathrm{m/s}$. (a) What is its initial acceleration if it is launched vertically? (b) Can it reach the first cosmic velocity if the mass of the shell and equipment is $m_0' = 60\ \mathrm{t}$?
A rocket with a structural mass $m_s = 2\ \mathrm{t}$ and fuel mass $m_z = 1.5\ \mathrm{t}$ burns fuel at $k = 250\ \mathrm{kg/s}$; the exhaust speed relative to the rocket is $u = 1\ \mathrm{km/s}$. (a) After what time does it reach $v = 200\ \mathrm{m/s}$? (b) What is its final speed? (Neglect gravity and air resistance.)
A spacecraft of initial mass $m_a$ moves with constant velocity $v_0$. To change its course, a side thruster ejects gas at a constant speed $u$ relative to the craft, perpendicular to the initial velocity, until the mass has dropped to $m_k$. By what angle is the trajectory deflected?
A body of mass $m = 2\ \mathrm{kg}$ is lifted vertically by a constant force $F$. While raising the body by $h = 1\ \mathrm{m}$, the force does work $A = 78.5\ \mathrm{J}$. Find the acceleration.
A constant force $F = 1\ \mathrm{N}$ gives a body an acceleration $a = 10\ \mathrm{cm/s^2}$. If the body starts from rest, what is its kinetic energy after $t = 5\ \mathrm{s}$?
To take off, an airplane of mass $m = 4\ \mathrm{t}$ must reach $v = 144\ \mathrm{km/h}$ at the end of a runway of length $s = 100\ \mathrm{m}$. The rolling friction coefficient is $\mu = 0.2$. Assuming uniform acceleration, what engine power is needed at take-off?
A car of mass $m = 4000\ \mathrm{kg}$ moves at $v_0 = 120\ \mathrm{km/h}$ on a horizontal road. The resistance force is $F_{\text{tr}} = 10\ \mathrm{kN}$. How far does the car travel after the engine is switched off?
When a shell is fired horizontally, the barrel of a cannon (mass $m_1 = 1000\ \mathrm{kg}$) recoils by $l = 0.25\ \mathrm{m}$. The shell has mass $m_2 = 40\ \mathrm{kg}$ and leaves at $v_0 = 800\ \mathrm{m/s}$. Find the average force of the recoil absorbers.
A cube-shaped container with edge $a$ is filled with water to half its height and stands with one face on a horizontal floor. How much work is needed to tip it over about one of its bottom edges? (Treat the container's own mass as negligible; its effect is added at the end.)
A body of mass $m$ starts sliding from the top of an inclined plane of height $h$. Find its kinetic energy at the start, halfway along the path and at the bottom. Neglect friction.
A tractor pulls a trailer at a constant $v_1 = 15\ \mathrm{km/h}$ on a road with friction coefficient $\mu_1 = 0.01$. It then drives onto a road with $\mu_2 = 0.1$. What is its speed there if its power output stays the same?
A construction crane lifts a load of $m = 1.2\ \mathrm{t}$ to a height $h = 18\ \mathrm{m}$ in $t = 40\ \mathrm{s}$. Friction consumes 3% of the motor's power. What power must the motor deliver?
A hydroelectric plant has a dam of height $h = 70\ \mathrm{m}$. At the bottom, the water leaves the turbine at $v = 22\ \mathrm{m/s}$. What is the efficiency of the plant (neglecting other losses)?
An electric motor (efficiency $\eta_1 = 0.95$) drives a pump (efficiency $\eta_2 = 0.90$) that pushes water up a pipe whose end is $h = 10\ \mathrm{m}$ above the pump. The flow rate is $Q = 200\ \mathrm{dm^3/s}$. What electrical power does the motor draw?
A train of mass $m = 400\ \mathrm{t}$ speeds up uniformly from $v_1 = 40\ \mathrm{km/h}$ to $v_2 = 60\ \mathrm{km/h}$ over a level distance $s = 1\ \mathrm{km}$. What power must the locomotive develop for this (neglecting friction)?
Two elastic balls of masses $m_1 = 10\ \mathrm{g}$ and $m_2 = 20\ \mathrm{g}$ hang side by side, just touching, on threads of equal length $l = 18\ \mathrm{cm}$. The ball $m_1$ is pulled aside until its thread makes $\alpha = 60^\circ$ with the vertical and released; it collides elastically with $m_2$. Find the largest angles the threads make after the collision.
A sailboat of mass $m$ is driven by a constant wind force, so that its position is $s = At^2 + Bt + C$ ($A$, $B$, $C$ constants). Find (a) the work done by the wind during the time $t$ from the start, (b) the instantaneous power of the wind. (Water resistance is neglected.)
The kinetic energy of a body moving on a circle of radius $R$ depends on the distance travelled as $E_k = ks^2$ ($k$ constant). How does the force acting on the body depend on $s$?
A body slides from the top of an inclined plane of height $h = 6\ \mathrm{m}$ and, after continuing along a horizontal path, stops at a point 25 m away horizontally from its starting point. The coefficient of friction is the same on the incline and on the horizontal path. Find it.
A sled starts from rest at point A of a slope, slides down to its foot B and continues along a horizontal stretch to C, where it stops. $AB = BC$, and the coefficient of friction $\mu = 0.1$ is the same along the whole path. Find the angle of the slope.
A car of mass $m = 2\ \mathrm{t}$ drives up a road that rises $h = 4\ \mathrm{m}$ per $l = 100\ \mathrm{m}$. The coefficient of friction (rolling resistance) is $\mu = 0.08$. Find the work and power of the engine if the car covers $s = 3\ \mathrm{km}$ in $t = 4\ \mathrm{min}$ at constant speed.
Two bodies of masses $m_1 = 2\ \mathrm{kg}$ and $m_2 = 4\ \mathrm{kg}$ hang on a rope over a massless, frictionless pulley. Initially the heavier one is $h = 2\ \mathrm{m}$ higher than the lighter one; they start from rest. Find the kinetic energy of the system when the two are level.
On a level road stretch of $s = 3\ \mathrm{km}$ a car of mass $m = 1.5\ \mathrm{t}$ speeds up from $v_1 = 36\ \mathrm{km/h}$ to $v_2 = 72\ \mathrm{km/h}$; the rolling friction coefficient is $\mu = 0.02$. Find (a) the engine's work on this stretch, (b) its average power (assuming uniform acceleration).
A body of mass $m$ is launched up an incline of angle $\theta$ with speed $v_0$; the coefficient of friction is $\mu$. (a) How far up does it go? (b) How much work is done against friction on the way?
A body of mass $m$ is held at a height $H$ above a table. Below it, standing on the table, is an undeformed spring of natural length $l_0$ and stiffness $k$. The body is released and falls onto the spring. What is the largest compression of the spring? What is the largest extension if the body sticks to the spring and oscillates?
A simple pendulum of length $l = 0.5\ \mathrm{m}$ is pulled aside by $\alpha = 60^\circ$, and the bob is given a speed $v_0 = 3\ \mathrm{m/s}$ downward, perpendicular to the string. Where does the bob get to? Neglect friction.
A metal ball of mass $m = 30\ \mathrm{kg}$ hangs on a rope of length $l = 10\ \mathrm{m}$. It is given a speed $v_0$ perpendicular to the rope, and the rope swings out to $\alpha = 60^\circ$. Find $v_0$ and the energy given to the ball.
A simple pendulum of length $l = 1\ \mathrm{m}$ is pulled aside by $\alpha = 60^\circ$ and given a speed $v_0$ downward, perpendicular to the string. What must $v_0$ be for the bob to go all the way round?
A bullet of mass $m = 20\ \mathrm{g}$ is fired at $v_0 = 1000\ \mathrm{m/s}$ and hits the ground at $v = 200\ \mathrm{m/s}$. How much energy was spent overcoming air resistance? (Neglect the difference in height.)
A body on a rope is whirled in a vertical circle, moving only under gravity and the rope (no driving force along the path). Prove that the greatest rope tension is at least 6 times the body's weight.
A man of mass $m_1 = 80\ \mathrm{kg}$, standing on rough ground, can throw a stone of mass $m_2 = 2\ \mathrm{kg}$ at $v_2 = 10\ \mathrm{m/s}$. (a) What speed can he give the stone standing on smooth ice? (b) What is then the stone's speed relative to the man? (c) Does he develop the same power in both cases?
A cart of mass $m$ starts from rest at height $h$ and runs along a track ABC that turns at B into a vertical semicircle of radius $R$ (B at the bottom, C at the top). Friction is negligible. (a) What is the smallest $h$ for which the cart stays on the track up to C? (b) How does the normal force depend on the position on the semicircle?
A cart of mass $m$ is pushed at A with speed $v_0$ along a track ABC (A on the level of B), which at B turns into a vertical semicircle of radius $R$. What is the smallest initial kinetic energy for the cart to get round to C without leaving the track, if 2% of the energy is lost to friction? What is its speed at C?
A track consists of two cylindrical arcs of equal radius $R$: a concave arc AB and a convex arc BC, joined smoothly at B with a horizontal tangent. A is $h$ above B, and C is $h$ below B. A small body starts from rest at A and slides without friction. What must $h$ be for the body to leave the track exactly at C?
A body of mass $m = 2\ \mathrm{kg}$ is thrown downward from a height $H = 200\ \mathrm{m}$ at $v_0 = 198\ \mathrm{m/s}$. It strikes sand and sinks $h = 0.5\ \mathrm{m}$ into it. Find the average resistance force of the sand (neglect air resistance).
A small body of mass $m$ slides without friction down a curved surface AB from height $h$ and at B moves onto a plank of mass $M$ lying on a smooth floor. Friction between the body and the plank eventually brings them to a common speed. How much energy is dissipated by friction?
A ski jumper starts with negligible speed from height $H$ and leaves the jump horizontally at height $h$ above the ground. What height $h$ of the take-off gives the longest jump (measured horizontally from the foot of the jump), and how long is it? Neglect friction and air resistance.
A small ball moves along a track and slides over a cylindrical surface BC of radius $R = 2\ \mathrm{m}$. At C, where the radius makes $\alpha = 45^\circ$ with the vertical, the ball leaves the surface. Find how far (horizontally) it flies.
Water enters a turbine at $v_1 = 4\ \mathrm{m/s}$ and leaves at $v_2 = 1\ \mathrm{m/s}$; the inlet is $h = 1.5\ \mathrm{m}$ above the outlet; the mass flow is $Q' = 300\ \mathrm{kg/s}$; the turbine's efficiency is $\eta = 0.85$. Find the power.
A water drop of mass $m = 0.1\ \mathrm{g}$ falls from rest from a height $H = 500\ \mathrm{m}$. It evaporates so that its mass decreases by $\Delta m = kt$, $k = 0.01\ \mathrm{g/s}$. At what height is its kinetic energy greatest? What fraction of its mass has it lost then, and what is that maximum kinetic energy? (Neglect air resistance; the evaporation does not push the drop.)
A car of mass $m = 1400\ \mathrm{kg}$ moves at a constant $v_1 = 40\ \mathrm{km/h}$. By how much must the engine's force increase for the car to double its speed over a distance $s = 280\ \mathrm{m}$? Neglect friction.
A car of mass $m = 1400\ \mathrm{kg}$ moves at a constant $v_1 = 40\ \mathrm{km/h}$. By how much must the engine's force increase for the car to double its speed over a distance $s = 280\ \mathrm{m}$? Neglect friction.
A body of mass $m = 2\ \mathrm{kg}$ is at the top of a smooth hemispherical dome of height (radius) $R = H = 10\ \mathrm{m}$. (a) What is the smallest horizontal speed $v_0$ with which it leaves the dome immediately, without sliding on its surface? (b) If it is given half that speed, where does it leave the dome?
A small body starts from rest at the top of a smooth hemisphere of radius $R$ standing on a horizontal floor. After leaving the hemisphere it hits the floor and bounces elastically (the speed does not change). (a) At what height does it leave the hemisphere? (b) How high does it rise after the bounce? (c) What is its speed at the highest point of that flight?
A body of mass $m_1 = 15\ \mathrm{kg}$ slides down a smooth incline of angle $\alpha = 60^\circ$ from a height $h = 10\ \mathrm{m}$ and at the bottom plunges into a cart full of sand of mass $m_2 = 90\ \mathrm{kg}$, standing on a smooth horizontal floor. With what speed do they move on together?
A body of mass $m$ is pushed slowly and uniformly up a curved surface by a force $\vec F$ that is always tangent to the path. Find the work of $\vec F$ between points 1 and 2, which differ in height by $h$ and are $l$ apart horizontally. The coefficient of friction is $\mu$.
A tram of mass $m$ moves so that its speed depends on the distance travelled as $v = ks^{3/2}$ ($k$ constant). Find (a) its acceleration, (b) the work of all forces acting on it.
A body of mass $m$ is thrown at angle $\alpha$ to the horizontal with speed $v_0$. Find the instantaneous power of the gravitational force as a function of time.
A point moves from $\vec r_1 = (1, 1, 1)\ \mathrm{m}$ to $\vec r_2 = (6, 6, 6)\ \mathrm{m}$ while a constant force $\vec F = 5\,\hat\imath\ \mathrm{N}$ acts on it. Find the work of this force.
A body of mass $m$, initially at rest, is acted on by the force $\vec F = At\,\hat\imath + Bt^2\,\hat\jmath$ ($A$, $B$ constants). Find the power of this force at time $t$.
A body of mass $m_1 = 4\ \mathrm{kg}$ moving at $v_1 = 10\ \mathrm{m/s}$ hits a body of twice its mass at rest; they stick together. By how much does the internal energy of the system increase?
Three balls of masses $m_1$, $m_2$, $m_3$ lie at rest along a line. The first is given speed $v_1$ and hits the second, which then hits the third; both collisions are elastic and head-on. What mass $m_2$ gives the third ball the greatest speed?
A hammer of mass $m_1 = 100\ \mathrm{kg}$ falls from a height $H = 3\ \mathrm{m}$ onto an anvil of mass $m_2 = 500\ \mathrm{kg}$ standing on an elastic (springy) base. After the impact the hammer and anvil move together. Find their speed just after the impact.
Two balls of equal mass move toward each other on a smooth horizontal surface at $v_1 = 10\ \mathrm{m/s}$ and $v_2 = 20\ \mathrm{m/s}$. What are their velocities after a head-on elastic collision?
A forging hammer of mass $m_1 = 1.2\ \mathrm{t}$ strikes a piece of iron lying on an anvil of mass $m_2 = 25\ \mathrm{t}$ at $v_1 = 5\ \mathrm{m/s}$. (a) How much energy goes into deforming the iron? (b) What is the efficiency of the blow?
A body of mass $m_1 = 2\ \mathrm{kg}$ moving at $v_1 = 3\ \mathrm{m/s}$ catches up with a body of mass $m_2 = 3\ \mathrm{kg}$ moving at $v_2 = 1\ \mathrm{m/s}$ along the same line in the same direction. Find their velocities after the collision if it is (a) perfectly inelastic, (b) elastic. What mass ratio $m_1/m_2$ would make the first body stop in an elastic collision?
A plastic ball falls from a height $h_1 = 1\ \mathrm{m}$ onto a horizontal plate and bounces. The time from release to the second impact is $t = 1\ \mathrm{s}$. Find the coefficient of restitution $k$ (ratio of the rebound speed to the impact speed).
A body of mass $m_1$ hits a body of mass $m_2$ at rest in a one-dimensional, perfectly inelastic collision. What fraction of the kinetic energy is turned into internal energy, as a function of $k = m_1/m_2$?
A body of mass $m_1$ collides elastically and head-on with a body of mass $m_2$ at rest. How much energy is passed to the second body? Express the ratio of the first body's initial energy to the transferred energy through $k = m_1/m_2$, and discuss $k = 1$.
A ball of mass $m$ makes head-on elastic collisions with bodies of mass $15m$ (each at rest before the collision). After how many collisions has its energy dropped by a factor of 100?
A metal ball falls freely from a height $h = 2\ \mathrm{m}$ onto a heavy metal plate and bounces back without losing speed. Find the time between two successive impacts.
Neutrons in a reactor must be slowed down. Using the result of Translational Dynamics 144, find which target slows a neutron best in a head-on elastic collision: (a) a proton ($m_p \approx m_n$), (b) an alpha particle ($m_\alpha \approx 4m_n$).
In a cloud chamber an alpha particle collides elastically with a nucleus of an unknown gas that is practically at rest. After the collision the alpha particle is deflected by $\theta_1 = 8^\circ 27'$ from its original direction, and the unknown particle moves off at $\theta_2 = 60^\circ$ to that direction. Find the mass ratio and identify the particle.