Gravitational Potential Energy: Practice Problem & Solution
Assertion (A): If the product of surface area and density is same for two planets, escape velocities at surface will be same for both planets. Reason (R): For given mass of a planet \( v_e \propto R^{-1/2} \)
Short Solution (Best for NEET):
To solve this problem, we apply the core principles of Gravitational Potential Energy. Understanding the underlying formula is key to arriving at the correct answer below:
Assertion (A) is true.
Escape velocity \( v_e = \sqrt{\frac{2GM}{R}} \). Substitute \( M = \rho \frac{4}{3} \pi R^3 \) to get \( v_e = \sqrt{\frac{8G\pi R^2 \rho}{3}} \). If \( R^2 \rho \) is constant (derived from \( A\rho \) being constant), then \( v_e \) is constant. Reason (R) is true, as \( v_e = \sqrt{\frac{2GM}{R}} \) shows \( v_e \propto R^{-1/2} \) for constant \( M \). However, (R) does not explain (A), as the conditions are different.
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