The coordinates of a moving particle at any time $t$ are given by $x = \alpha t^3$ and $y = \beta t^3$. The speed of the particle at time $t$ is given by

The position coordinates of a particle moving in a $3-D$ coordinate system are given by $x = a \cos \omega t$,$y = a \sin \omega t$,and $z = a \omega t$. The speed of the particle is:

Which two motions are considered to be combined for motion in a plane?

$A$ particle initially at the origin starts moving in the $xy$-plane with a velocity component $\vec{V} = (6 + 2t) \hat{i} + (4 + 2\sqrt{3}t) \hat{j} \text{ m/s}$. The acceleration of the particle in $\text{m/s}^2$ is ($x, y$ are measured in meters,$t$ in seconds,respectively).

The position vector of a moving body at any instant of time is given as $\overrightarrow{r} = (5t^2 \hat{i} - 5t \hat{j}) \text{ m}$. The magnitude and direction of velocity at $t = 2 \text{ s}$ is,

The magnitude of acceleration and velocity of a particle moving in a plane,whose position vector is $\vec{r} = 3t^2 \hat{i} + 2t \hat{j} + \hat{k}$ at $t = 2 \text{ s}$,are,respectively: