A uniform solid spherical ball is rolling down a smooth inclined plane from a height h. The velocity attained by the ball when it reaches the bottom of the inclined plane is v. If ball when it reaches the bottom of the inclined plane is v. If the ball is now thrown vertically upwards with the same velocity `v`, the maximum height to which the ball will rise is

Options

A. `(5h)/(8)`
B. `(3h)/(5)`
C. `(5h)/(7)`
D. `(7h)/(9)`

Answer

C

Explanation

Total kinetic energy of a body which is rolling without slipping is given by
`K_("total") = K_("rot")+K_("trans")`
For solid spherical ball,
`I=(2)/(5)mR^(2) and v=R omega`(along the diameter)
where, R is radius of spherical ball
So, `K_("total")=(1)/(2)((2)/(5)mR^(2))omega^(2)+(1)/(2)mR^(2)omega^(2)=(7)/(10)mR^(2)omega^(2)=(7)/(10)mv^(2)`
Potential energy = Kinetic energy
`mgh = (7)/(10)mv^(2) rArr v^(2)=(10)/(7)gh` ...(i)
From vertical projection,
`v^(2)=u^(2)+2gh'rArr(10)/(7)gh=0+2gh'rArr h'=(5)/(7)h`.