$A$ body is projected with velocity $u$ making an angle $\alpha$ with the horizontal. Its velocity when it is perpendicular to the initial velocity vector $u$ is

$A$ particle is projected from the ground at an angle $\theta$ with the horizontal with speed $u$. The ratio of the radius of curvature of its trajectory at the point of projection to the radius of curvature at the maximum height is ........

$A$ point moves in the $x-y$ plane according to the law $x = 3 \cos 4t$ and $y = 3(1 - \sin 4t)$. The distance travelled by the particle in $2 \text{ s}$ is...........$m$ (where $x$ and $y$ are in meters).

Two particles move on a circular path (one just inside and the other just outside) with angular velocities $\omega$ and $5\omega$ starting from the same point. Then

The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R} = 4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}$,where $R$ is in meters,$t$ is in seconds,and $\hat{i}$ and $\hat{j}$ denote unit vectors along $x$- and $y$-directions,respectively. Which one of the following statements is wrong for the motion of the particle?

The figure shows the $(x, t)$ and $(y, t)$ diagrams of a particle moving in $2$ dimensions. If the particle has a mass of $500 \, g$,find the force (magnitude and direction) acting on the particle.