A tennis ball of mass $m$ is projected vertically upwards with initial speed $u_{0}$ and reaches its highest point at time $T$. In addition to uniform gravity, the ball experiences air resistance, which produces a frictional force of magnitude $\alpha v$, where $v$ is the ball's speed and $\alpha$ is a positive constant. Show by dimensional analysis that $T$ can be written in the form

$$T=\frac{m}{\alpha} f(\lambda)$$

for some function $f$ of a dimensionless quantity $\lambda$.

Use the equation of motion of the ball to find $f(\lambda)$.