Two particles are projected from the same point with the same speed at different angles $\theta _1$ and $\theta _2$ to the horizontal. They have the same range. Their times of flight are $t_1$ and $t_2$ respectively.

Show that for a projectile the angle between the velocity and the $x$ -axis as a function of time is given by

$\theta(t)=\tan ^{-1}\left(\frac{v_{0 y}-g t}{v_{0 x}}\right)$

Show that the projection angle $\theta_{0}$ for a projectile launched from the origin is given by

$\theta_{0}=\tan ^{-1}\left(\frac{4 h_{m}}{R}\right)$

Where the symbols have their usual meaning.

A ball is thrown from a roof top at an angle of $45^o$ above the horizontal. It hits the ground a few seconds later. At what point during its motion, does the ball have $(a)$ greatest speed $(b)$ smallest speed $(c)$ greatest acceleration – Explain.

The speed of a projectile at its maximum height is $\frac {\sqrt 3}{2}$ times its initial speed. If the range of the projectile is $P$ times the maximum height attained by it, $P$ is equal to

Galileo writes that for angles of projection of a projectile at angles $(45 + \theta )$ and $(45 – \theta )$, the horizontal ranges described by the projectile are in the ratio of (if $\theta \le 45)$