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 moves such that its position vector $\overrightarrow{\mathrm{r}}(\mathrm{t})=\cos \omega \mathrm{t} \hat{\mathrm{i}}+\sin \omega \mathrm{t} \hat{\mathrm{j}}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\overrightarrow{\mathrm{v}}(\mathrm{t})$ and acceleration $\overrightarrow{\mathrm{a}}(\mathrm{t})$ of the particle

A particle is moving eastwards with a speed of $6 \,m / s$. After $6 \,s$, the particle is found to be moving with same speed in a direction $60^{\circ}$ north of east. The magnitude of average acceleration in this interval of time is ....... $m / s ^2$

“Explain average acceleration and instantaneous acceleration.”

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