Q.

A satellite of mass ‘m’  is orbiting a planet of mass M in a circular orbit of radius R. The satellite changes its orbit by expelling small amount of fuel [5% of its existing mass ] with high speed U in a direction opposite to the direction of its orbital velocity. Find Umax (approximately) of the expelled mass for which, satellite cannot escape the gravitational field of planet.

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a

GMR1/2

b

5GMR1/2

c

7GMR1/2

d

3GMR1/2

answer is C.

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Detailed Solution

V0=GMR Binding energy =GMm2RIf Ve is escape speed, then12mVe2=12mV02+GMm2R⇒Ve=2GMR1/2=2V0Using conversation of momentum;mV0=(m−Δm)Ve−ΔmUmax⇒mV0=m−5100m2V0−5m100Umax Umax=(2×95−100)5V0≈7V0
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