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Show that the velocity of an object rele...

Show that the velocity of an object released from rest at infinity reaches the earth with a velocity equal to the escape velocity.

Text Solution

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By the principle of conservation of energy for the object.
TE at infinity = TE at the surface of the earth
(KE + PE) at infinity = (KE + PE) at the surface of the earth.
As the body is released from rest at infinity, its KE at infinity is zero and its PE at infinity is also zero. If v is its velocity on reaching the earth its KE becomes equal t `(1)/(2)mv^(2)` and `PE = (GMm)/(R)`
`therefore 0 + 0 = (1)/(2) mv^(2) -(GMm)/(R) , (1)/(2)v^(2) = (Gm)/(R)`
The velocity of the object, `V = sqrt((2GM)/(R)) = V_(e)`.
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