Two projectiles are fired with the same speed, one at α_1 =…
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.
Explanation of Method
Comparing throws at 45° and 30°.
Step 1: Formulas for a throw at angle $\alpha$
\[ H = \frac{v_0^2\sin^2\alpha}{2g}, \]
\[ R = \frac{v_0^2\sin 2\alpha}{g}, \]
\[ T = \frac{2v_0\sin\alpha}{g}. \]
With the same $v_0$, everything except the angle cancels in the ratios.
Step 2: (a) Heights
\[ \frac{H_1}{H_2} = \frac{\sin^2 45^\circ}{\sin^2 30^\circ} = \frac{1/2}{1/4}. \]
\[H_1 : H_2 = 2\]
Step 3: (b) Ranges
\[ \frac{R_1}{R_2} = \frac{\sin 90^\circ}{\sin 60^\circ} = \frac{2}{\sqrt3}. \]
\[R_1 : R_2 \approx 1.15\]
Step 4: (c) Flight times
\[ \frac{T_1}{T_2} = \frac{\sin 45^\circ}{\sin 30^\circ} = \sqrt2. \]
\[T_1 : T_2 \approx 1.41\]
- 45°
- 30°
Result
\[\boxed{\,H_1 : H_2 = 2\,}\]
\[\boxed{\,R_1 : R_2 \approx 1.15\,}\]
\[\boxed{\,T_1 : T_2 \approx 1.41\,}\]
Remark
Note. 45° gives the largest possible range for a given speed on level ground. A 60° throw would have the same range as the 30° one (complementary angles), but would fly higher and longer.