Question

A body is projected vertically from Earth's surface with \( \left(\frac{1}{3}\right)^{\text{rd}} \) of the escape velocity. The maximum height reached by the body is (\( R \) = radius of Earth).

• A. $\frac{\text{R}}{4}$
• B. $\frac{\text{R}}{8}$
• C. $\frac{R}{9}$
• D. $\frac{R}{6}$

Hint:

For $v = v_e/n$, the maximum height is $h = \frac{R}{n^2 - 1}$. Here $h = \frac{R}{3^2 - 1} = \frac{R}{8}$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the conservation of mechanical energy: $\frac{1}{2}mv^2 - \frac{GMm}{R} = -\frac{GMm}{R+h}$.

Step 2: Meaning
Escape velocity $v_e = \sqrt{\frac{2GM}{R}}$. Given $v = \frac{1}{3}v_e$, so $v^2 = \frac{1}{9} \times \frac{2GM}{R} = \frac{2GM}{9R}$.

Step 3: Analysis
$\frac{1}{2}m(\frac{2GM}{9R}) - \frac{GMm}{R} = -\frac{GMm}{R+h}$
$\frac{GMm}{9R} - \frac{GMm}{R} = -\frac{GMm}{R+h} \implies \frac{1}{9R} - \frac{1}{R} = -\frac{1}{R+h}$
$-\frac{8}{9R} = -\frac{1}{R+h} \implies 8R + 8h = 9R \implies 8h = R \implies h = R/8$.

Step 4: Conclusion
The maximum height is $R/8$. Final Answer: (B)