Two bodies of mass $10 \, kg$ and $5 \, kg$ are moving in concentric orbits of radii $R$ and $r$ such that their periods are the same. What is the ratio of their centripetal accelerations?

$A$ particle leaves the origin with an initial velocity $\vec{v} = (3 \hat{i}) \text{ m s}^{-1}$ and a constant acceleration $\vec{a} = (-1 \hat{i} - 0.5 \hat{j}) \text{ m s}^{-2}$. The position vector of the particle, when it reaches its maximum $x$-coordinate, is:

$A$ body of mass $1 \, kg$ starts moving from rest at $t = 0$ in a circular path of radius $8 \, m$. Its kinetic energy varies as a function of time as: $K.E. = 2t^2 \, J$,where $t$ is in seconds. Then:

$A$ particle is projected from the ground with velocity $u$ at an angle $\theta$ from the horizontal. Match the following two columns.

Column $I$	Column $II$
$(A)$ Average velocity between initial and final points	$(p)$ $u \sin \theta$
$(B)$ Change in velocity between initial and final points	$(q)$ $u \cos \theta$
$(C)$ Change in velocity between initial and highest points	$(r)$ Zero
$(D)$ Average velocity between initial and highest points	$(s)$ None of the above

$A$ 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.

$A$ sportsman runs around a circular track of radius $r$ such that he traverses the path $ABAB$. The distance travelled and displacement,respectively,are