Answer: A) 18 m
Solution:
The height a man can jump is inversely proportional to the acceleration due to gravity \( g \):
\[
h \propto \frac{1}{g}
\]
Acceleration due to gravity \( g \) is given by:
\[
g = \frac{4}{3} \pi G R \rho
\]
Where:
- \( R \) = Radius of the planet
- \( \rho \) = Density of the planet
Let \( g_e \) be the acceleration due to gravity on Earth and \( g_p \) on the planet.
The new \( g_p \) is proportional to the product of density \( \rho \) and radius \( R \):
\[
g_p \propto R_p \rho_p
\]
Given:
- \( R_p = \frac{1}{3} R_e \)
- \( \rho_p = \frac{1}{4} \rho_e \)
Thus:
\[
g_p \propto \left( \frac{1}{3} \right) \left( \frac{1}{4} \right) = \frac{1}{12} g_e
\]
This means the gravity on the planet is \( \frac{1}{12} \) of that on Earth.
Since height \( h \) is inversely proportional to \( g \):
\[
h_p = h_e \times \frac{g_e}{g_p}
\]
Substitute the values:
\[
h_p = 1.5 \times \frac{g_e}{\frac{1}{12} g_e} = 1.5 \times 12 = 18 \, \text{m}
\]
Final Answer: The man can jump approximately 18 m on the planet.
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