A particle is projected at angle $\theta$ with horizontal from ground. The slop $(m)$ of the trajectory of the particle varies with time $(t)$ as ...........

The time of flight of an object projected with speed $20 \,ms ^{-1}$ at an angle $30^{\circ}$ with the horizontal, is .... $s$

A projectile is launched from the origin in the $xy$ plane ( $x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance. $r =\sqrt{ x ^2+ y ^2}$ from the origin is plotted against $x$, the resulting curves show different behaviours for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure below. For $\alpha_1, r ( x )$ keeps increasing with $x$ while for $\alpha_2$, $r(x)$ increases and reaches a maximum, then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at an angle $\alpha_c\left(\alpha_1 < \alpha_c < \alpha_2\right)$. The value of $\alpha_c$ is [ignore where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity]

A projectile is thrown upward with a velocity $v_0$ at an angle $\alpha$ to the horizontal. The change in velocity of the projectile when it strikes the same horizontal plane is

In a circus, a performer throws an apple towards a hoop held at $45 \,m$ height by another performer standing on a high platform (see figure). The thrower aims for the hoop and throws the apple with a speed of $24 \,m / s$. At the exact moment that the thrower releases the apple, the other performer drops the hoop. The hoop falls straight down. At ............ $m$ height above the ground does the apple go through the hoop?

Ratio between maximum range and square of time of flight in projectile motion is