What is the smallest speed with which an object must be…

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.

Explanation of Method

Smallest speed for a 64 m throw.

Step 1: Best angle

For a given speed, the range $R = \dfrac{v_0^2\sin 2\alpha}{g}$ is largest when $\sin 2\alpha = 1$, i.e. at $\alpha = 45^\circ$. So the smallest speed for a given range is reached at 45°:

\[ D = \frac{v_0^2}{g} \;\Longrightarrow\; v_0 = \sqrt{Dg} = \sqrt{64 \cdot 9.81}. \]

\[v_{0,\min} \approx 25.1\ \mathrm{m/s} \ \ (\text{at } 45^\circ)\]

$D = 64\ \mathrm{m}$ $45^\circ$ $20^\circ,\ 70^\circ:\ 41\ \mathrm{m}$

Throws with the same speed (25.1 m/s), drawn to scale. Only the 45° throw reaches 64 m; any other angle falls short, so 45° needs the least speed.

Result

\[\boxed{\,v_{0,\min} \approx 25.1\ \mathrm{m/s} \ \ (\text{at } 45^\circ)\,}\]

Remark

Note. With this speed, throws at 20° or 70° land only about 41 m away. The highest point of the 45° throw is $D/4 = 16\ \mathrm{m}$.