The figure shows a sphere of mass $500 \ g$ moving in a steady flow of air in the $x$-direction. The air-stream exerts an essentially constant force of $0.9 \ N$ on the sphere in the $x$-direction. If at $t = 0$ the sphere is moving as shown in the figure,determine the time required for the sphere to cross the $y$-axis again in $\sec$. (in $.66$)

$A$ body is moving with velocity $30 \; m/s$ towards east. After $10 \; s$,its velocity becomes $40 \; m/s$ towards north. The average acceleration of the body is ...... $m/s^2$.

$A$ particle of mass $m$ is projected at $45^o$ at $V_0$ speed from point $P$ at $t = 0$. The angular momentum of the particle about $P$ at $t = \frac{V_0}{g}$ 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	$(s)$ $4 \sin 2$

In a circus,a performer throws an apple towards a hoop held at $45 \, m$ height by another performer standing on a high platform (see figure). The thrower aims for the hoop and throws the apple with a speed of $24 \, m/s$. At the exact moment that the thrower releases the apple,the other performer drops the hoop. The hoop falls straight down. At what height above the ground does the apple go through the hoop (in $, m$)?

$A$ particle is moving in the $xy$-plane and crosses the origin at time $t=0$. The equation of motion of the particle is $y=4x^2$. If the velocity of the particle is $\vec{v}=(2\hat{i}+2\hat{j}) \text{ m s}^{-1}$ and acceleration is $\vec{a}=(a\hat{j}) \text{ m s}^{-2}$, then the magnitude of $a$ is