Question

The maximum vertical distance through which a full dressed astronaut can jump on the Earth is 0.5 m. Estimate the maximum vertical distance through which he can jump on the moon, which has a mean density \[\dfrac{2}{3}rd\] that of the Earth and radius one quarter that of the Earth.
A. 1.5m
B. 3m
C. 6m
D. 7.5m

Answer 1

Hint: Planets and stars have their properties such as their size, mass and density. These properties determine the gravitational pull of a planet. Moon is a star of the Earth and is smaller in size than the Earth. The density of the moon is 60% as that of the Earth. Therefore the gravitational potential energy on the Moon is less than that on the Earth.

Complete step-by-step answer:
Step I:
Let the astronaut jump with an initial velocity ‘u’ and a final speed of ‘v’. Let ‘h’ be the maximum height he jumps to and ‘g’ be acceleration due to gravity.

Step II:
Then by equation of motion
\[{v^2} = {u^2} - 2gh\]

Step III:
If the jumping speed on the Earth and moon is same then
\[h \propto \dfrac{1}{g}\]
And the gravitational acceleration on a planet\[g = \dfrac{{GM}}{{{R^2}}}\]---(i)
M is mass of Earth and \[M = \dfrac{4}{3}\pi \rho {R^3}\]---(ii)

Step IV:
Substituting value of M from equation (ii) to (i)
\[g = \dfrac{{G\dfrac{4}{3}\rho \pi {R^3}}}{{{R^2}}}\]
Where G= Gravitational constant
R=Radius and \[\rho = \]density
\[g = \dfrac{4}{3}\pi \rho GR\]
Value of \[\dfrac{4}{3}\pi G\] is constant.
Therefore, \[g \propto \rho R\].

Step V:
Value of ‘g’ on moon is, \[{g_m} = \dfrac{{{g_e}}}{6}\]
Since astronauts have energy to jump only 0.5m on Earth. Therefore on moon,
\[P.E{._{moon}} = P.E{._{Earth}}\]
\[m{g_m}{h_m} = m{g_e}{h_e}\]
\[{g_m}{h_m} = {g_e}{h_e}\]
\[{h_m} = \dfrac{{{g_e}}}{{{g_m}}}{h_e}\]
\[{h_m} = 6 \times {h_e}\]
\[{h_m} = 6 \times \dfrac{1}{2}\]
\[{h_m} = 3m\]

Step VI:
Maximum vertical distance he can jump on moon is = 3m
Option B is the right answer.

Note: Whenever a person jumps, he tries to move out of the influence of gravitational force. But since the gravitational force on the moon is 17% that of Earth. So using the same force as that of Earth, one can jump even much higher on the moon than that on the Earth.