$A$ ball of mass $160 \, g$ is thrown up at an angle of $60^{\circ}$ to the horizontal at a speed of $10 \, m/s$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly $\left(g=10 \, m/s^{2}\right)$ (in $kg \cdot m^{2}/s$).

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

$A$ projectile fired at $30^{\circ}$ to the ground is observed to be at the same height at time $t_1 = 3 \, s$ and $t_2 = 5 \, s$ after projection,during its flight. The speed of projection of the projectile is $......... \, m \, s^{-1}$ (Given $g = 10 \, m \, s^{-2}$).

The second's hand of a watch has $6\, cm$ length. The speed of its tip and the magnitude of the difference in velocities of its tip at any two perpendicular positions will be respectively:

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 ............

The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R} = 4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}$,where $R$ is in meters,$t$ is in seconds,and $\hat{i}$ and $\hat{j}$ denote unit vectors along $x$- and $y$-directions,respectively. Which one of the following statements is wrong for the motion of the particle?