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Question

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 v1 and that of m2 is v2)

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A
v1=v2
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B
v1=m22Gd(m1+m2),v2=m12Gd(m1+m2)
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C
v1=m12Gd(m1+m2),v2=m22Gd(m1+m2)
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D
v1=m22Gm1,v2=m22Gm2
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Solution

The correct option is B v1=m22Gd(m1+m2),v2=m12Gd(m1+m2)
Solution:
We choose the reference point, infinity, where the total energy of the system is zero.
So, initial energy of the system = 0
Final energy =12m1v21+12m2v22Gm1m2d

From conservation of energy,Initial energy = Final energy
0=12m1v21+12m1v22=Gm1m2d...(1)

By conservation of linear momentum
m1v1+m1v2=0

or v1v2=m2m1v2=m1m2v1

Putting value of v2 in equation , we get

m1v2+m2(m1v1m2)2=2Gm1m2d

m1m2v21+m21v21m2=2Gm1m2d

v1=2Gm22d(m1+m2)=m22Gd(m1+m2)
Similarly v2=m12Gd(m1+m2)


Hence B is the correct option

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