164- In the figure below, a pendulum ball is released from point $A$ and passes through the lowest point of the path with speed $V$. When the ball's speed reaches $\dfrac{\sqrt{2}}{2}\ V$, what angle does the string make with the vertical?
$$\left(g = 10\ \frac{\text{m}}{\text{s}^2},\ \cos 53^\circ = 0.6\right)$$
[Figure: Pendulum of length $1\ \text{m}$ released from point $A$ at $53^\circ$ from vertical]
(Air resistance is neglected.)

• [(1)] $60$ (2) $45$
• [(3)] $37$ (4) $30$

A tennis ball (treated as hollow spherical shell) starting from O rolls down a hill. At point A the ball becomes air borne leaving at an angle of $30^{\circ}$ with the horizontal. The ball strikes the ground at B. What is the value of the distance AB? (Moment of inertia of a spherical shell of mass $m$ and radius $R$ about its diameter $= \frac{2}{3}mR^2$)
(1) 1.87 m
(2) 2.08 m
(3) 1.57 m
(4) 1.77 m

If the angle of elevation of a cloud from a point $P$ which is $25 m$ above a lake be $30 ^ { \circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60 ^ { \circ }$, then the height of the cloud (in meters) from the surface of the lake is :
(1) 50
(2) 60
(3) 45
(4) 42

A swimmer can swim with velocity of $12 \mathrm {~km} / \mathrm { h }$ in still water. Water flowing in a river has velocity $6 \mathrm {~km} / \mathrm { h }$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is $\_\_\_\_$. (Round off to the Nearest Integer) (find the angle in degree)