Question

An astronaut on earth can throw a ball straight up to a height of $15m.$ How high can he throw the ball on Mars?

Short answer

The astronaut can throw a ball straight up to a height of $39.7m$on Mars.

Step-by-step solution

Given information

Height on earth's surface is $h=15\mathrm{m}.$

Free-fall acceleration on earth's surface: $g_e=9.81\mathrm{m}/{\mathrm{s}}^2.$

Free-fall acceleration on Mars's surface: $g_m=3.71\mathrm{m}/{\mathrm{s}}^2.$

Calculation

Let the mass of the ball be m.

Hence, the energy required to throw the ball to height $h_e$is:

$E_e=m{g}_e{h}_e.$____(1)

Consider, he can throw the same ball to the height $h_m$on Mars's surface.

Hence, energy required on Mars to throw the ball is :$E_m=m{g}_m{h}_m.$_____(2)

Continuation of calculation

Since, the person is same, the amount of energy he can supply to the ball is same.

Hence, compare the energies from the equation (1) and (2) :

$$\begin{gathered} m{g}_e{h}_e=m{g}_m{h}_m \\ \Rightarrow h_m=\frac{g_{ϵ}{h}_e}{g_m}.\_\_\_\_\_\_\left(3\right) \end{gathered}$$

Continuation of calculation

Now substitute the values to the equation (3) calculate the height.

$$\begin{gathered} h_m=\frac{\left(9.81\mathrm{m}/{\mathrm{s}}^2\right)(15\mathrm{m})}{\left(3.71\mathrm{m}/{\mathrm{s}}^2\right)} \\ =39.7\mathrm{m}. \end{gathered}$$

Final answer

The astronaut can throw a ball straight up to a height of $39.7m$on Mars.