$A$ particle is moving in a circle of radius $R$ with constant speed $V$. The magnitude of average acceleration after half revolution is

$A$ particle moves in space according to the equation $\vec{r} = (\sin t \hat{i} + \cos t \hat{j} + t \hat{k}) \text{ m}$. Find the time $t$ when the position vector and acceleration vector are perpendicular to each other.

$A$ particle moves in a plane along an elliptic path given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. At point $(0, b)$,the $x$-component of velocity is $u$. The $y$-component of acceleration at this point is

At any time $t$,the coordinates of a moving particle are $x = at^2$ and $y = bt^2$. The speed of the particle is

$A$ particle is projected from the ground with velocity $u$ at an angle $\theta$ with the horizontal. The horizontal range,maximum height,and time of flight are $R, H$,and $T$ respectively. They are given by $R = \frac{u^2 \sin 2\theta}{g}$,$H = \frac{u^2 \sin^2 \theta}{2g}$,and $T = \frac{2u \sin \theta}{g}$. Now,keeping $u$ fixed,$\theta$ is varied from $30^o$ to $60^o$. Then,

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 ........ $m$.