For angles of projection of a projectile at angles $(45^\circ + \theta)$ and $(45^\circ - \theta)$,the horizontal range described by the projectile are in the ratio of:

$A$ fighter jet is flying horizontally at a certain altitude with a speed of $200 \; m/s$. When it passes directly overhead an anti-aircraft gun,a bullet is fired from the gun at an angle $\theta$ with the horizontal to hit the jet. If the bullet speed is $400 \; m/s$,the value of $\theta$ will be $\dots \; ^\circ$.

$A$ projectile is fired at an angle of $45^{\circ}$ with the horizontal. The elevation angle of the projectile at its highest point as seen from the point of projection is:

$A$ ball is thrown at $30^{\circ}$ with the horizontal from the top of a roof $20 \,m$ high with a speed of $13 \,ms^{-1}$. At what distance from the throwing point will the ball, once again, be at a height of $20 \,m$ from the ground (in $\,m$)? $(g = 10 \,ms^{-2})$

$A$ missile is fired for maximum range at your town from a place $100 \, km$ away from you. If the missile is first detected at its half-way point,how much warning time will you have? (Take $g = 10 \, m/s^2$)

In the given figure for a projectile,find the relation for $y$ in terms of $x_1, x_2$ and $\theta$.