Question

If air resistance in neglected, show (algebraically) that a ball thrown vertically upwards with a speed of Vo will have the same speed, Vo, when it comes back down to starting point.

Answer 1

Velocity =   displacement/time ; In variable form, we can say
 Vo = d/t ....(1)
Using the kinematic equationd =  t*(Vo + Vf)/2 ...(2)
Where Vf is the final velocity, but in this scenario,  we are saying it is the same as the initial velocity hence, (2) becomes

d = t*(Vo + Vo)/2
d = t*(2*Vo)/2
the 2's cancel
d = t*Vo
solving for Vo we get,
Vo = d/t which is the exact same as 1. Keep in mind the distance traveled does not change

Answer 2

Explanation:

If the air resistance is negligible then in that case we can say that there will be only one force on the ball

Force of gravity

so here we will have

$F_g = mg$

so the acceleration of the ball is given as

$a = g$

now we have

$v_f^2 - v_i^2 = 2a s$

as the ball returns to initial position

so the displacement must be zero

$v_f^2 - v_i^2 = 0$

so here we have

$v_f = vi$