question_answer

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 \[{{m}_{1}}\] is \[{{v}_{1}}\] and that of \[{{m}_{2}}\] is \[{{v}_{2}}\])          [JEE ONLINE 12-04-2014]

A)  \[{{v}_{1}}={{v}_{2}}\]        

B)  \[{{v}_{1}}={{m}_{2}}\sqrt{\frac{2G}{d\left( {{m}_{1}}+{{m}_{2}} \right)}}\] \[{{v}_{2}}={{m}_{1}}\sqrt{\frac{2G}{d\left( {{m}_{1}}+{{m}_{2}} \right)}}\]

C)  \[{{v}_{1}}={{m}_{1}}\sqrt{\frac{2G}{d\left({{m}_{1}}+{{m}_{2}}\right)}}\]\[{{v}_{2}}={{m}_{2}}\sqrt{\frac{2G}{d\left( {{m}_{1}}+{{m}_{2}} \right)}}\]

D)  \[{{v}_{2}}={{m}_{2}}\sqrt{\frac{2G}{{{m}_{1}}}}\]            \[{{v}_{2}}={{m}_{2}}\sqrt{\frac{2G}{{{m}_{2}}}}\]

Correct Answer: B

Solution :

[b] We choose reference point, infinity, where total energy of the system is zero.

So, initial energy of the system = 0

Final energy

From conservation of energy,

Initial energy = Final energy

or                    .(1)

By conservation of linear momentum

or

Putting value of in equation (1), we get

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