A point `P` lies on the axis of a fixed ring of mass `M` and radius `a`, at a distance `a` from its centre `C`. A small particle starts from `P` and reaches `C` under gravitational attraction only. Its speed at `C` will be.

Options

A. `sqrt((2Gm)/(r) ( sqrt(2) - 1))`
B. `sqrt((Gm)/(r))`
C. `sqrt((2Gm)/(r) (1 - (1)/(sqrt(2)))`
D. `sqrt((2Gm)/(r))`

Answer

C

Explanation

Let .M. be the mass of the particle
Now, `E_("initial") = E_("final")`
i.e., `(GMm)/(sqrt(2)r) + 0 = ("GMm")/(r) + (1)/(2) MV^(2)`
or, `(1)/(2) MV^(2) = ("GMm")/(r) [ 1 - (1)/(sqrt(2)) ] `
`rArr (1)/(2) V^(2) = (Gm)/(r) [ 1 - (1)/(sqrt(2)) ] or, V = sqrt((2Gm)/(r) (1 - (1)/(sqrt(2))))`