Consider a point $P$ on the circumference of a disc rolling along a horizontal surface. If $R$ is the radius of the disc, the distance through which $P$ moves in one full rotation of the disc is

The position of a particle moving along $x$-axis given by $x=\left(-2 t^3+3 t^2+5\right) m$. The acceleration of particle at the instant its velocity becomes zero is ....... $m / s ^2$

A particle starts from rest and performing circular motion of constant radius with speed given by $v = \alpha \sqrt x$ where $\alpha$ is a constant and $x$ is the distance covered. The correct graph of magnitude of its tangential acceleration $(a_t)$ and centripetal acceleration $(a_c)$ versus $t$ will be:

A particle initially at rest is subjected to two forces. One is constant, the other is a retarding force proportional to the particle velocity. In the subsequent motion of the particle :

The trajectory of a projectile in a vertical plane is $y =\alpha x -\beta x ^{2},$ where $\alpha$ and $\beta$ are constants and $x \& y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection $\theta$ and the maximum height attained $H$ are respectively given by :-