Two hypothetical planets of masses m1 and m2 are at rest when they… | JEE Main 2014 (12 Apr, Shift 1)

Q29JEE Main 2014 · 12 APR · SHIFT 1Difficult

Gravitational Potential & Energy

Two hypothetical planets of masses m1 and m2 are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is 'd'?

(Speed of m1 is $v$ 1 and that of m2 is $v$ 2)

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Answer: (b)

Momentum conservation (start from rest): $m_1v_1 = m_2v_2$
Energy: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{Gm_1m_2}{d}$
Put $v_2 = \frac{m_1v_1}{m_2}$ : $\frac{1}{2}m_1v_1^2\cdot\frac{m_1+m_2}{m_2} = \frac{Gm_1m_2}{d}$
$v_1 = m_2\sqrt{\frac{2G}{d(m_1+m_2)}}$ and similarly $v_2 = m_1\sqrt{\frac{2G}{d(m_1+m_2)}}$
So the answer is (b).

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