For the given system,find the value of $\theta_2$.

State whether the following statements are true or false:
$(a)$ Momentum and the change in momentum are always in the same direction.
$(b)$ Action force and reaction force always act simultaneously and on the same body.
$(c)$ The maximum static frictional force depends on the area of the contact surface.

Give the magnitude and direction of the net force acting on a stone of mass $0.1\; kg$,
$(a)$ just after it is dropped from the window of a stationary train,
$(b)$ just after it is dropped from the window of a train running at a constant velocity of $36\; km/h$,
$(c)$ just after it is dropped from the window of a train accelerating with $1\; m/s^2$,
$(d)$ lying on the floor of a train which is accelerating with $1\; m/s^2$,the stone being at rest relative to the train.
Neglect air resistance throughout.

$A$ truck starting from rest moves with an acceleration of $5 \ m/s^2$ for $1 \ s$ and then moves with constant velocity. The velocity $v/s$ time graph for the block on the truck with respect to the ground is: (Assume that the block does not fall off the truck and the coefficient of friction $\mu = 0.2$,$g = 10 \ m/s^2$)

$A$ 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. . . . .

$A$ person of mass $60\, kg$ is inside a lift of mass $940\, kg$ and presses the button on the control panel. The lift starts moving upwards with an acceleration of $1.0\, m/s^2$. If $g = 10\, m/s^2$,the tension in the supporting cable is .......... $N$.