Projectile Motion on Earth, Moon, and Mars

The same throw on three different worlds gives wildly different results — and there's a simple equation that ties it all together. This is why Apollo astronauts hit golf balls so far on the Moon and why dust kicked up by Mars rovers takes so long to settle.

The Physics

For an identical launch from level ground, the range, height, and flight time are:

$R = \dfrac{v_0^{2}\sin(2\theta)}{g} \qquad H = \dfrac{v_0^{2}\sin^{2}\theta}{2g} \qquad T = \dfrac{2 v_0\sin\theta}{g}$

Range scales inversely with gravity. Halving $g$ doubles the range; cutting $g$ to one-sixth (the Moon) gives six times the range. Time of flight also scales as $1/g$, while peak height scales as $1/g$.

Step-by-Step Solution

Given:

• Initial speed: $v_0 = 20\;\text{m/s}$
• Launch angle: $\theta = 45°$
• Earth gravity: $g_E = 9.81\;\text{m/s}^{2}$
• Mars gravity: $g_M = 3.71\;\text{m/s}^{2}$
• Moon gravity: $g_L = 1.62\;\text{m/s}^{2}$

Find: Range, peak height, and flight time on each world.

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Step 1 — Compute the vertical launch component (same on all worlds).

$v_{0y} = v_0\sin 45° = 20 \times \dfrac{\sqrt{2}}{2} \approx 14.142\;\text{m/s}$

This is the same on every world because the launch is the same — it's only gravity that differs.

Step 2 — On Earth ($g = 9.81$).

$T_E = \dfrac{2 v_0\sin\theta}{g_E} = \dfrac{2(14.142)}{9.81} = \dfrac{28.284}{9.81} \approx 2.884\;\text{s}$

$H_E = \dfrac{v_{0y}^{2}}{2 g_E} = \dfrac{(14.142)^{2}}{19.62} = \dfrac{200}{19.62} \approx 10.19\;\text{m}$

$R_E = \dfrac{v_0^{2}\sin(90°)}{g_E} = \dfrac{400}{9.81} \approx 40.77\;\text{m}$

Step 3 — On Mars ($g = 3.71$).

$T_M = \dfrac{28.284}{3.71} \approx 7.624\;\text{s}$

$H_M = \dfrac{200}{2(3.71)} = \dfrac{200}{7.42} \approx 26.95\;\text{m}$

$R_M = \dfrac{400}{3.71} \approx 107.82\;\text{m}$

Step 4 — On the Moon ($g = 1.62$).

$T_L = \dfrac{28.284}{1.62} \approx 17.46\;\text{s}$

$H_L = \dfrac{200}{2(1.62)} = \dfrac{200}{3.24} \approx 61.73\;\text{m}$

$R_L = \dfrac{400}{1.62} \approx 246.91\;\text{m}$

Step 5 — Compare the ratios.

Mars range relative to Earth: $107.82 / 40.77 \approx 2.65$ — the Mars throw goes 2.6× farther.

Moon range relative to Earth: $246.91 / 40.77 \approx 6.05$ — the Moon throw goes 6× farther.

These match the inverse gravity ratios: $9.81/3.71 \approx 2.65$ and $9.81/1.62 \approx 6.05$. ✓

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Answer:

• Earth ($g = 9.81$): $R \approx 40.77\;\text{m}$, $H \approx 10.19\;\text{m}$, $T \approx 2.88\;\text{s}$
• Mars ($g = 3.71$): $R \approx 107.82\;\text{m}$, $H \approx 26.95\;\text{m}$, $T \approx 7.62\;\text{s}$
• Moon ($g = 1.62$): $R \approx 246.91\;\text{m}$, $H \approx 61.73\;\text{m}$, $T \approx 17.46\;\text{s}$

The Moon throw stays in the air for nearly 17 seconds and travels almost a quarter of a kilometer — from the same modest 20 m/s kick that only manages 40 m on Earth.

Why the Moon Is So Generous

Apollo 14 commander Alan Shepard famously hit a couple of golf balls on the Moon's surface, claiming they went "for miles and miles." With gravity ~1/6 of Earth's, a club swing that goes 250 m on Earth would (in vacuum, ignoring drag) go ~1500 m on the Moon. Range scales linearly with $1/g$.

Try It

• All three balls launch at the same instant — watch them separate dramatically as the Moon ball stretches far beyond.
• Try increasing the launch angle toward 80° — the Moon ball reaches breathtaking heights.
• Drop the speed: even small kicks travel surprisingly far on the Moon.