Question

The escape velocity of a body on the earth's surface is \(v_{e}\). A body is thrown with a speed of \(\sqrt{6}v_{e}\). Assume that the sun and the planets do not influence the motion of the body, the velocity of a body at infinite distance is \(v_{\infty }\). The value of \(\frac{v_{\infty }}{v_{e}}\) is

• A. \(\sqrt{6}\)
• B. \(2\)
• C. \(\sqrt{5}\)
• D. \(3\)

Hint:

Apply energy conservation between the surface and infinity, and use the escape speed to remove the potential energy term.

Answer 1

The Correct Option is C

Step 1: Formula-first approach
The speed at infinity is given by $v_{\infty}=\sqrt{v^{2}-v_e^{2}}$.

Step 2: Substitute
$v_{\infty}=\sqrt{(\sqrt{6}v_e)^{2}-v_e^{2}}=\sqrt{5}\,v_e$.

Step 3: Match
The ratio is $\sqrt{5}$, option (C).

Final Answer:
The ratio is $\sqrt{5}$. \[ \boxed{\text{(C) }\sqrt{5}} \]