The escape velocity of a body on the earth's surface is v e . A body is thrown with a speed of 6 v e . Assume that the sun and the planets do not influence the motion of the body, the veloc

Question

The escape velocity of a body on the earth's surface is v e ​ . A body is thrown with a speed of 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 ∞ ​ . The value of v e ​ v ∞ ​ ​ is

TestHub · Accepted Answer

By the principle of conservation of mechanical energy:

$K_{i} + U_{i} = K_{f} + U_{f}$

$\frac{1}{2} m v^{2} - \frac{GM m}{R} = \frac{1}{2} m v_{\infty}^{2} + 0$

Since the escape velocity is given by $v_{e} = \frac{2 GM}{R}$ , we have $\frac{GM m}{R} = \frac{1}{2} m v_{e}^{2}$ .

Substituting this into the energy equation:

$\frac{1}{2} m v^{2} - \frac{1}{2} m v_{e}^{2} = \frac{1}{2} m v_{\infty}^{2}$

$v_{\infty}^{2} = v^{2} - v_{e}^{2}$

Given $v = 6 v_{e}$ , we get:

$v_{\infty}^{2} = (6 v_{e})^{2} - v_{e}^{2}$

$v_{\infty}^{2} = 6 v_{e}^{2} - v_{e}^{2} = 5 v_{e}^{2}$

$v_{\infty} = 5 v_{e}$

$\frac{v _{\infty}}{v _{e}} = 5$

Answer: $5$

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