$A$ body of mass $1\,kg$ is projected with velocity $50\,m/s$ at an angle of $30^{\circ}$ with the horizontal. At the highest point of its path,a force of $10\,N$ starts acting on the body for $5\,s$ vertically upward in addition to the gravitational force. What is the horizontal range of the body? $\left(g=10\,m/s^2\right)$

$A$ particle moves over a $xy$ plane with a constant acceleration $\vec{a} = (4.0 \, m \, s^{-2}) \hat{i} + (4.0 \, m \, s^{-2}) \hat{j}$. At time $t = 0$, the velocity is $\vec{v}_0 = (4.0 \, m \, s^{-1}) \hat{i}$. The speed of the particle when it is displaced by $6.0 \, m$ parallel to the $x$-axis is,

$A$ car moving with a certain velocity jumps from an inclined plane placed at one bank of a river and reaches the other bank by attaining a maximum height of $80 \,m$. If the same car, moving with the same velocity, jumps from another inclined plane having a different angle of inclination and reaches the same point on the other bank by attaining a maximum height of $45 \,m$, then the width of the river is: (in $\,m$)

$A$ particle of mass $m$ moves along a circular path of radius $r$ with a time-dependent centripetal acceleration $a_c = k^2rt^2$,where $k$ is a constant. What is the power delivered to the particle by the force acting on it?

Two balls are thrown with the same velocity,one vertically upward and the other at an angle of $60^\circ$ with the vertical. What is the ratio of their potential energies at their respective maximum heights?

$A$ particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution. If this particle were projected with the same speed at an angle $\theta$ to the horizontal,the maximum height attained by it equals $4R$. The angle of projection,$\theta$,is then given by