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Question

A small body starts falling onto the Sun from a distance equal to the radius of the Earth. The initial velocity of the body is equal to zero in the heliocentric reference frame. If the time taken by the body to fall into the sun is 6.45×10k. Then k is

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Solution

Consider that the body has velocity v when it is at distance x from the sun.
By conservation of mechanical energy,

12mv2+(Gmmx)=0Gmmr

v=2gmr×(vx)x

=dxdt=2gmr×(vx)x

todt=r2Gmr0xrxdx

substituting, x=rsin2θ

t=r2gm×(πr2)

t=(28)T

T is the time period of revolution of earth

t=64.5 days

t=6.45×101

So K=1

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