A body A of mass m is moving in a circular or | vedclass

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Question

(A) Initially,the body of mass $m$ is moving in a circular orbit of radius $R$. So,it must be moving with orbital speed $v_{0} = \sqrt{\frac{GM}{R}}$.

Vedclass · Accepted Answer

starts moving in an elliptical orbit around the planet

Vedclass · Answer

(A) Initially,the body of mass $m$ is moving in a circular orbit of radius $R$. So,it must be moving with orbital speed $v_{0} = \sqrt{\frac{GM}{R}}$.After the collision,let the combined mass move with speed $v_{1}$.Using the conservation of linear momentum:$m v_{0} + \frac{m}{2} \left(\frac{v_{0}}{2}ight) = \left(m + \frac{m}{2}ight) v_{1}$$m v_{0} + \frac{m v_{0}}{4} = \frac{3m}{2} v_{1}$$\frac{5m v_{0}}{4} = \frac{3m}{2} v_{1}$$v_{1} = \frac{5}{4} 	imes \frac{2}{3} v_{0} = \frac{5}{6} v_{0}$.Since the speed after the collision $(v_{1} = \frac{5}{6} v_{0})$ is not equal to the orbital speed $(v_{0})$,the motion cannot be circular.Since the velocity remains tangential,it cannot fall vertically towards the planet.Also,the speed after the collision is less than the escape speed $(v_{e} = \sqrt{2} v_{0})$,so the body cannot escape the gravitational field.Therefore,the combined body will move in an elliptical orbit around the planet.

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