A ball is dropped vertically from `a` height `d` above the ground . It hits the ground and bounces up vertically to a height ` (d)//(2). Neglecting subsequent motion and air resistance , its velocity `v` varies with the height `h` above the ground as

Explanation

O to A: Velocity of the ball at any height h above the ground
`v^(2)=0+2g(d-h)`
`h=d, v=0`
`h=0, v=v_(0)=sqrt(2 gd)`
Shape: `y^(2)=2g(d-x)`(parabola, open leftward)
Velocity is vertically downward and hence, negative.
A to B: `v^(2)=v_(1)^(2)-2gh`
`0=v_(1)^(2)-2g d/2implies v_(1)=sqrt(gd)`
`h=0, v=v_(1)=sqrt(gd)`
`h=d/2, v=0`
`v^(2)=v_(1)^(2)-2gh=gh-2gx`
Shape: `y^(2)=gd-2gx`(parabola, open leftward)
Velocity is vertically upward and hence, positive.