$A$ cart moves with a constant speed along a horizontal circular path. From the cart,a particle is thrown up vertically with respect to the cart.

An athlete completes one round of a circular track of radius $10 \, m$ in $40 \, sec$. The distance covered by him in $2 \, min \ 20 \, sec$ is ........ $m$.

$A$ particle of mass $M$ is moving in a horizontal circle of radius $R$ with uniform speed $v$. When the particle moves from one point to a diametrically opposite point,its

The given diagram shows a straight length of tape being wound on to a reel. The reel is rotating about a fixed axis with constant angular velocity and its radius is increasing at a steady rate. Which graph correctly shows the variation with time of the speed $v$ at which the tape moves towards the roll?

$A$ projectile is launched from point $A$ of the given landscape with a water body as shown in the diagram. The launching angle is $15^{\circ}$. From the following, identify the right initial velocity of the projectile with which it will fall somewhere in between the points $C$ and $D$. [Assume $g = 10 \,m/s^2$] (in $\,m/s$)

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