A rocket is launched at α = 70^° to the horizontal and…

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).

Explanation of Method

Initial speed and range from the time to the top.

Step 1: Initial speed

At the top the vertical velocity is zero: $v_0\sin\alpha - g t_m = 0$, so

\[ v_0 = \frac{g t_m}{\sin\alpha} = \frac{9.81 \cdot 80}{0.9397} = \frac{784.8}{0.9397}. \]

\[v_0 \approx 835\ \mathrm{m/s}\]

Step 2: Range

The flight is symmetric, so the total time is $2t_m = 160\ \mathrm{s}$: $R = v_0\cos\alpha \cdot 2t_m = 835.2 \cdot 0.3420 \cdot 160$.

\[R \approx 45\,700\ \mathrm{m} \approx 45.7\ \mathrm{km}\]

Check with the range formula

$ R = \tfrac{v_0^2\sin 140^\circ}{g} = \tfrac{835.17^2 \cdot 0.6428}{9.81} \approx 45\,703\ \mathrm{m}\ \checkmark $ (the original 45 690 m is a rounding difference)

$\text{top at } t_m = 80\ \mathrm{s}$ $R \approx 45.7\ \mathrm{km}$ $70^\circ$ $H \approx 31.4\ \mathrm{km}$

Drawn to scale (no air resistance). The rocket spends 80 s rising and 80 s falling, landing about 45.7 km away.

Result

\[\boxed{\,v_0 \approx 835\ \mathrm{m/s}\,}\]

\[\boxed{\,R \approx 45\,700\ \mathrm{m} \approx 45.7\ \mathrm{km}\,}\]