Two hypothetical planets of masses m_1 and m_ | vedclass

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

Question

(B) The initial total energy of the system at an infinite distance is $0$.By the law of conservation of energy,the total energy at separation $d$ must

Vedclass · Accepted Answer

$v_1 = m_2 \sqrt{\frac{2G}{d(m_1 + m_2)}}, v_2 = m_1 \sqrt{\frac{2G}{d(m_1 + m_2)}}$

Vedclass · Answer

(B) The initial total energy of the system at an infinite distance is $0$.By the law of conservation of energy,the total energy at separation $d$ must also be $0$:$\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 - \frac{G m_1 m_2}{d} = 0$$\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{G m_1 m_2}{d} \quad ... (i)$By the law of conservation of linear momentum (as no external force acts on the system):$m_1 v_1 - m_2 v_2 = 0 \implies v_2 = \frac{m_1}{m_2} v_1$Substituting $v_2$ into equation $(i)$:$\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 \left( \frac{m_1}{m_2} v_1 \right)^2 = \frac{G m_1 m_2}{d}$$\frac{1}{2}m_1 v_1^2 \left( 1 + \frac{m_1}{m_2} \right) = \frac{G m_1 m_2}{d}$$\frac{1}{2} v_1^2 \left( \frac{m_1 + m_2}{m_2} \right) = \frac{G m_2}{d}$$v_1^2 = \frac{2 G m_2^2}{d(m_1 + m_2)} \implies v_1 = m_2 \sqrt{\frac{2G}{d(m_1 + m_2)}}$Similarly,$v_2 = m_1 \sqrt{\frac{2G}{d(m_1 + m_2)}}$.

Similar Questions

$A$ body of mass $m$ is released from a height of $2R$ above the Earth's surface. What will be its kinetic energy at a height of $R$ from the Earth's surface? (Where $R$ is the radius of the Earth)

What is the change in the gravitational potential energy of an object when it moves away from the Earth and when it comes closer to the Earth?

Two masses $m_1$ and $m_2$ are at rest at infinity. Find their relative velocity of approach due to gravitational attraction when their separation is $d$.

$A$ body of mass $m$ falls from a height $R$ above the surface of the earth,where $R$ is the radius of the earth. What is the velocity attained by the body on reaching the ground? (Acceleration due to gravity on the surface of the earth is $g$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module