Vertical motion

velocity versus time for the solid metal ball

The vertical velocity in metres per second versus the instant of time in seconds after launching (at t=0) the solid metal ball with an initial upward speed of v0=20 m/s.

For the velocity as a function of time of a vertically moving solid metal ball close to the surface of earth we have learned in secondary school the expression

v(t) = dy(t)/dt = v₀ - gt

The instant t_max that the metal ball passes through the maximum height can be calculated from

0 = v₀ - gt_max

which has the solution:

t_max = v₀/g

The maximum height itself follows by substitution of t_max in the expression for the height

y_max = v₀t_max - gt_max²/2 = v₀²/2g