Derive the equations of motion for a particle (a) falling vertically (b) projected vertically.

Explanation

A body filling from a height h : Let us consider an object of mass m falling from a height 'h . Let us assume there is no air resistance, The object experiences acceleration 'g' due to gravity that is constant.
a = g
If the particle is thrown with initial velocity 'u' downward that is in negative - Y-aixs, then velocity and position of the particle at any time t is given by
v = u + gt . . . (1)
`y = ut +(1)/(2) g t^(2)" ". . . (2)`
When it is at a distance y from the hill -top , is
`v^(2) = u^(2) +2gy " ". . . (3)`
Suppose the particle starts from rest.
u = 0
For a point y from the hill -top
v = gt
` y = (1)/(2) g t^(2)`
`v^(2) = 2 gy`