A particle of mass m is released from infinit | vedclass

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Question

(C) The total mechanical energy of the particle is conserved. At infinity,the potential energy is $U_{\infty} = 0$ and the kinetic energy is $K_{\inft

Vedclass · Accepted Answer

$\sqrt {\frac{{3GM}}{R}} $

Vedclass · Answer

(C) The total mechanical energy of the particle is conserved. At infinity,the potential energy is $U_{\infty} = 0$ and the kinetic energy is $K_{\infty} = 0$. At the centre of the sphere,the potential energy $U_{c}$ is given by the formula for the potential inside a uniform solid sphere: $U_{c} = m 	imes V_{c} = m 	imes \left( -\frac{3GM}{2R} ight) = -\frac{3GMm}{2R}$. Applying the law of conservation of energy: $K_{\infty} + U_{\infty} = K_{c} + U_{c}$. $0 + 0 = \frac{1}{2}mV^{2} - \frac{3GMm}{2R}$. $\frac{1}{2}mV^{2} = \frac{3GMm}{2R}$. $V^{2} = \frac{3GM}{R}$. $V = \sqrt{\frac{3GM}{R}}$.

$A$ particle of mass $m$ is released from infinity and it moves towards a large solid sphere of mass $M$ and radius $R$ made of a jelly-like material. Neglecting the resistance offered by the material of the sphere,find the velocity of the particle when it crosses the centre of the sphere.

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