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:

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ}$ and $48^{\circ}$ with the horizontal are $R_{1}, R_{2}$ and $H_{1}, H_{2}$ respectively. Choose the correct option:

Two balls are thrown as shown in the figure and they reach the ground in the same time. What is the ratio of the maximum heights attained by them?

$A$ particle moving in the $xy$ plane experiences a velocity-dependent force $\overrightarrow{F} = k(v_y \hat{i} + v_x \hat{j})$,where $v_x$ and $v_y$ are the $x$ and $y$ components of its velocity $\overrightarrow{v}$. If $\overrightarrow{a}$ is the acceleration of the particle,then which of the following statements is true for the particle?

$A$ horizontal plane supports a stationary vertical cylinder of radius $R = 1 \ m$ and a disc $A$ attached to the cylinder by a horizontal thread $AB$ of length $l_0 = 2 \ m$ (seen in figure,top view). An initial velocity $v_0 = 1 \ m/s$ is imparted to the disc as shown in the figure. How many seconds will it move along the plane until it strikes against the cylinder? (All surfaces are assumed to be smooth.)

$A$ particle has an initial velocity $(2\hat i + 3\hat j)$ and an acceleration $(0.3\hat i + 0.2\hat j)$. Its speed after $10\,s$ is: