$A$ particle is moving in a circular path. The acceleration and momentum of the particle at a certain moment are $\vec{a} = (4\hat{i} + 3\hat{j}) \, m/s^2$ and $\vec{P} = (8\hat{i} - 6\hat{j}) \, kg \cdot m/s$. The motion of the particle is:

$A$ ball of mass $m$ is thrown vertically upward. Another ball of mass $2m$ is thrown at an angle $\theta$ with the vertical. Both the balls stay in the air for the same period of time. The ratio of the maximum heights attained by the two balls respectively is $\frac{1}{x}$. The value of $x$ is $.....$

$A$ particle is rotating in a circle of radius $1\,m$ with constant speed $4\,m/s$. In time $1\,s$,match the following (in $SI$ units) columns.

Column $I$	Column $II$
$(A)$ Displacement	$(p)$ $8 \sin 2$
$(B)$ Distance	$(q)$ $4$
$(C)$ Average velocity	$(r)$ $2 \sin 2$
$(D)$ Average acceleration	$(s)$ $4 \sin 2$

$A$ particle $A$ is projected vertically upwards. Another particle $B$ is projected at an angle of $45^{\circ}$. Both reach the same maximum height. The ratio of the initial kinetic energy of $A$ to that of $B$ is

$A$ person walks in such a way that he covers equal distance in each step. The person takes $2$ steps forward towards East, then takes a right turn and walks $4$ steps towards South, then takes a right turn and walks $6$ steps towards West, and then takes a right turn and walks further. The direction of his final position after a total of $20$ steps walk with respect to his initial position is:

$A$ smooth circular groove has a smooth vertical wall as shown in the figure. $A$ block of mass $m$ moves against the wall with a speed $v$. Which of the following curves represents the correct relation between the normal reaction $(N)$ on the block by the wall and the speed of the block $(v)$?