$A$ particle performs uniform circular motion in a horizontal plane. The radius of the circle is $20 \, cm$. The centripetal force acting on the particle is $10 \, N$. Its kinetic energy is ........ $J$.

The given diagram shows a straight length of tape being wound on to a reel. The reel is rotating about a fixed axis with constant angular velocity and its radius is increasing at a steady rate. Which graph correctly shows the variation with time of the speed $v$ at which the tape moves towards the roll?

$A$ projectile is launched from the origin in the $xy$ plane ($x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance $r = \sqrt{x^2 + y^2}$ from the origin is plotted against $x$,the resulting curves show different behaviors for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure. For $\alpha_1$,$r(x)$ keeps increasing with $x$,while for $\alpha_2$,$r(x)$ increases and reaches a maximum,then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at a critical angle $\alpha_c$ (where $\alpha_1 < \alpha_c < \alpha_2$). The value of $\alpha_c$ is (where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity).

$A$ ball is projected from the ground at an angle of $\theta$ from the horizontal. The graph of kinetic energy $(K)$ versus time $(t)$ will be:

The coordinates of a particle moving in the $x-y$ plane are given by: $x = 2 + 4t$,$y = 3t + 8t^2$. The motion of the particle is:

$A$ charged particle of mass $m = 2 \ kg$ and charge $q = 1 \ \mu C$ is projected from a horizontal ground at an angle $\theta = 45^{\circ}$ with speed $u = 10 \ ms^{-1}$. In space,a horizontal electric field $E = 2 \times 10^7 \ NC^{-1}$ exists in the direction of projection. The range of the projectile is......$m$.