Suppose two planets (spherical in shape) of r | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) To reach the surface of the second planet,the satellite must cross the point where the net gravitational field is zero.Let the distance of this po

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) To reach the surface of the second planet,the satellite must cross the point where the net gravitational field is zero.Let the distance of this point from the centre of the planet of mass $M$ be $x$.$\frac{GM}{x^2} = \frac{G(9M)}{(8R-x)^2}$$\frac{1}{x} = \frac{3}{8R-x}$$8R-x = 3x \Rightarrow x = 2R$.Now,apply the law of conservation of energy between the surface of the first planet and the point $x = 2R$ (where the velocity is minimum,i.e.,$v_{min} = 0$):$E_{initial} = E_{final}$$\frac{1}{2}mv^2 - \frac{GMm}{R} - \frac{G(9M)m}{7R} = 0 - \frac{GMm}{2R} - \frac{G(9M)m}{6R}$$\frac{1}{2}v^2 = \frac{GM}{R} + \frac{9GM}{7R} - \frac{GM}{2R} - \frac{9GM}{6R}$$\frac{1}{2}v^2 = \frac{GM}{R} [1 + \frac{9}{7} - \frac{1}{2} - \frac{3}{2}]$$\frac{1}{2}v^2 = \frac{GM}{R} [1 + \frac{9}{7} - 2] = \frac{GM}{R} [\frac{9}{7} - 1] = \frac{GM}{R} [\frac{2}{7}]$$v^2 = \frac{4}{7} \frac{GM}{R}$$v = \sqrt{\frac{4}{7} \frac{GM}{R}}$Comparing this with $\sqrt{\frac{a}{7} \frac{GM}{R}}$,we get $a = 4$.

Similar Questions

Given that the mass of the Earth is $M$ and its radius is $R$. $A$ body is dropped from a height equal to the radius of the Earth above the surface of the Earth. When it reaches the ground,its velocity will be:

$A$ body (mass $m$) starts its motion from rest from a point distant $R_0$ $(R_0 > R)$ from the centre of the Earth. The velocity acquired by the body when it reaches the surface of the Earth will be ($G =$ universal constant of gravitation,$M =$ mass of Earth,$R =$ radius of Earth).

$A$ body of mass $m$ is lifted up from the surface of the earth to a height three times the radius of the earth. The change in potential energy of the body is (where $g$ is acceleration due to gravity at the surface of the earth).

$A$ body which is initially at rest at a height $R$ above the surface of the earth of radius $R$,falls freely towards the earth. Its velocity on reaching the surface of the earth is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module