$A$ body of mass $3 \, kg$ hits a wall at an angle of $60^\circ$ with the wall and returns at the same angle. The speed of the body is $10 \, m/s$ and the impact time is $0.2 \, s$. Calculate the force exerted on the wall.

Consider two blocks of mass $m_1 = 5 \, kg$ and $m_2 = 10 \, kg$ placed on a horizontal surface. The coefficient of static friction between the blocks and the surface is $\mu = 0.2$. $A$ horizontal force $F$ is applied to the $10 \, kg$ block. What is the maximum force $F$ (in $N$) that can be applied such that neither block moves? (Take $g = 10 \, m/s^2$)

Consider a system of blocks $A$, $B$, and $X$ as shown in the figure. The blocks $A$ and $B$ have equal mass $m$ and are connected by a massless string through a massless pulley. The coefficient of friction between block $A$ and $X$, and between block $B$ and $X$, is $\mu = 0.5$. If block $X$ moves on the horizontal frictionless surface, what should be its minimum acceleration $a$ such that blocks $A$ and $B$ remain stationary relative to $X$? ($g =$ acceleration due to gravity.)

For the given system,find the value of ${\theta _1}$ in degrees.

$A$ projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \text{ m/s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,$0.5 \text{ s}$ after the splitting. The other part,$t$ seconds after the splitting,falls to the ground at a distance $x$ meters from the point $O$. The acceleration due to gravity $g = 10 \text{ m/s}^2$.
$(1)$ The value of $t$ is. . . . . .
$(2)$ The value of $x$ is. . . . .

On a pulley of mass $M$ hangs a rope with two masses $m_{1}$ and $m_{2}$ $(m_{1} > m_{2})$ tied at the ends as shown in the figure. The pulley rotates without any friction,whereas the friction between the rope and the pulley is large enough to prevent any slipping. Which of the following plots best represents the difference between the tensions in the rope on the two sides of the pulley as a function of the mass of the pulley?