$A$ particle is projected with velocity $u$ from point $M$ on a frictionless incline of length $20\sqrt{2} \, m$. If it crosses a $40 \, m$ wide well at an angle of $45^o$,what should be its velocity at point $M$?

$A$ particle of mass $m$ is executing uniform circular motion on a path of radius $r$. If $p$ is the magnitude of its linear momentum,then the radial force acting on the particle is:

Show that for a projectile,the angle between the velocity and the $x$-axis as a function of time is given by $\theta(t) = \tan^{-1}\left(\frac{v_{0y} - gt}{v_{0x}}\right)$.
Show that the projection angle $\theta_0$ for a projectile launched from the origin is given by $\theta_0 = \tan^{-1}\left(\frac{4h_m}{R}\right)$,where the symbols have their usual meanings.

$A$ particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{n u^2}{25 g}$,where the value of $n$ is (Given $g$ is the acceleration due to gravity).

An insect trapped in a circular groove of radius $12 \; cm$ moves along the groove steadily and completes $7$ revolutions in $100 \; s$.
$(a)$ What is the angular speed,and the linear speed of the motion?
$(b)$ Is the acceleration vector a constant vector? What is its magnitude?

The position vector of a particle is given as $\vec{r} = (t^2 - 8t + 12)\hat{i} + t^2\hat{j}$. The time after which the velocity vector and acceleration vector become perpendicular to each other is equal to ........ $sec$.