The escape velocity for a planet is `v_(e)`. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the centre of the planet, its speed will be

Options

A. `v_(e )`
B. `(v_(e ))/(sqrt(2))`
C. `(v_(e ))/(2)`
D. zero

Answer

B

Explanation

From energy conservation
Potential inside sphere `V=-(GM)/(2R^(3))(3R^(2)-r^(2))`
`V_(s)=-(GM)/(R ), V_(0)=-(3GM)/(2R)`
Loss of potential energy of partical = Gain of kinetic energy
`m(V_(s)-V_(0))=(1)/(2)mv^(2)`
`v^(2)=2[-(GM)/(R )(-(3GM)/(R ))]rArr v= sqrt((GM)/(R )) rArr v = sqrt((2GM)/(2R)) rArr v = (V_(e ))/(sqrt(2))`