Question

How many times higher could an astronaut jump on the Moon than on Earth if his take-off speed is the same in both locations (gravitational acceleration on the Moon is about 1/6 of gon Earth)?

Short answer

The astronaut jumps six times higher on Moon compared to that on earth.

Step-by-step solution

Height of jump on Moon and earth

The height covered by the body is h. The velocity of the body at which he is jumping is V m/s. The gravitational acceleration of the earth is $\mathbf{g}{\mathbf{\text{\hspace{0.17em}m/s}}}^{\mathbf{\text{2}}}$.

The gravitational acceleration of the Moon is ${\mathbf{g}}_m\mathbf{m}/{\mathbf{s}}^2$.

$\mathbf{h}\mathbf{=}\frac{{\mathbf{v}}^{\mathbf{2}}}{\mathbf{2}\mathbf{g}}$

For Moon, the height of the jump is

${\mathbf{h}}_{\mathbf{M}}\mathbf{=}\frac{{{\mathbf{v}}_{\mathbf{M}}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{g}}_{\mathbf{M}}}$

Comparision of height

Now, as in the question, it is clearly mentioned that the gravitational acceleration of the Moon will be 1/6^th of that of the earth.

Hence it can write as:

${\mathbf{g}}_{\mathbf{M}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}}{\mathbf{g}}_{\mathbf{e}}$

Hence if we put the value of gm in the above equation, we get

$\begin{array}{l}{\mathrm{h}}_{\mathrm{M}}=\frac{{\mathrm{v}}^2}{2\times \left(\frac{1}{6}\mathrm{g}\right)}\\ {\mathrm{h}}_{\mathrm{M}}=6\left(\frac{{\mathrm{v}}^2}{2\times \left(\mathrm{g}\right)}\right)\\ {\mathrm{h}}_{\mathrm{M}}=6\mathrm{h}\end{array}$

Hence, it is clear that the jump height is higher in Moon cars than in Earth cars.