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 $\theta \le 45)$

The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by $y=\left(4 t-2 t^2\right) m$ and $x =(3 t)$ metre, where $t$ is in second and point of projection is taken as origin. The angle of projection of projectile with vertical is .........

A gun can fire shells with maximum speed $v_0$ and the maximum horizontal range that can be achieved is $R_{max} = \frac {v_0^2}{g}$.  If a target farther away by distance $\Delta x$ (beyond $R$) has to be hit with the same gun, show that it could be achieved by raising the gun to a height at least $h = \Delta x\,\left[ {1 + \frac{{\Delta x}}{R}} \right]$.

From the top of a tower of height $40\, m$, a ball is projected upwards with a speed of $20\, m/s$ at an angle $30^o$ to the horizontal. The ball will hit the ground in time ......... $\sec$ (Take $g = 10\, m/s^2$)

A stone is just released from the window of a train moving along a horizontal straight track. The stone will hit the ground following

A projectile thrown with velocity $v$ making angle $\theta$ with vertical gains maximum height $H$ in the time for which the projectile remains in air, the time period is