$A$ particle moves in a circle of radius $R$,with a constant speed $v$. Then,during a time interval $t = \pi R / 3v$,which of the following is true?

An aircraft executes a horizontal loop with a speed of $150\, m/s$ with its wings banked at an angle of $12^\circ$. The radius of the loop is .......... $km$ $(g = 10\, m/s^2 \text{ and } \tan 12^\circ = 0.2125)$

$A$ particle is projected. If after $2 \, s$ it makes an angle of $30^\circ$ with the horizontal and after $1 \, s$ more it is moving horizontally,find the initial velocity and angle of projection of the particle.

The position vector of a particle moving in a plane is given by $r = a \cos \omega t \hat{i} + b \sin \omega t \hat{j}$, where $\hat{i}$ and $\hat{j}$ are the unit vectors along the rectangular axes $X$ and $Y$; $a$, $b$, and $\omega$ are constants, and $t$ is time. The acceleration of the particle is directed along the vector:

Fill in the blanks:
$(a)$ If $\overrightarrow A \cdot \overrightarrow B = AB$,then the angle between $\overrightarrow A$ and $\overrightarrow B$ is ............
$(b)$ The velocity of a projectile at its maximum height is ............ (Take the angle of projection as $\theta$).
$(c)$ The projection of $\widehat i - 2\widehat j + 4\widehat k$ on the $y$-axis is ............

$A$ ball is projected from a point with a speed $V_0$ at a certain angle $\theta$ with the horizontal. From the same point and at the same instant,a person starts running with a constant speed $0.5 V_0$ to catch the ball. If the person catches the ball after some time,then the angle of projection of the ball is (in $^{\circ}$)