$A$ projectile crosses two walls of equal height $H$ symmetrically as shown. The height of each wall is ........ $m$.

$A$ 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 behaviors for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure. 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 a critical angle $\alpha_c$ (where $\alpha_1 < \alpha_c < \alpha_2$). The value of $\alpha_c$ is (where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity).

$A$ ball is projected from the ground at an angle of $\theta$ from the horizontal. The graph of kinetic energy $(K)$ versus time $(t)$ will be:

State whether the following statements are true or false:
$(a)$ If the angular velocity is constant on a circular path,the linear velocity is also constant.
$(b)$ In projectile motion,the velocity vector of the projectile is always perpendicular to the acceleration vector.
$(c)$ For the maximum horizontal range $R$ of a projectile,the maximum height attained is $\frac{R}{4}$.
$(d)$ If $|\vec{A} \times \vec{B}| = AB$,then the angle between $\vec{A}$ and $\vec{B}$ is zero.

$A$ spherical bob of mass $250 \ g$ is attached to the end of a string having length $50 \ cm$. The bob is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is $72 \ N$. The maximum possible value of angular velocity of the bob (in $rad/s$) is

$A$ stone is projected with a velocity $20 \sqrt{2} \, m/s$ at an angle of $45^{\circ}$ to the horizontal. The average velocity of the stone during its motion from the starting point to its maximum height is $.......... \, m/s$ (take $g = 10 \, m/s^2$).