$A$ ball is dropped from height $5m$. The time after which ball stops rebounding if coefficient of restitution between ball and ground $e = 1/2$, is .................. $\mathrm{sec}$

A particle is moved from $(0, 0)$ to $(a, a)$ under a force $\vec F = (3\hat i + 4\hat j)$ from two paths. Path $1$ is $OP$ and path $2$ is $OQP$. Let $W_1$ and $W_2$ be the work done by this force in these two paths respectively. Then

A force acts on a $3.0\ g$ particle in such a way that the position of the particle as a function of time is given by:

$x = 3t - 4t^2 + t^3$ 

Where $x$ is in metres and $t$ is in seconds. The work done during the first $4\ s$ is ................. $\mathrm{mJ}$

Two identical $5\,\,kg.$  blocks are moving with same speed of $2\,\,m/s$  towards each other along a frictionless horizontal surface. The two blocks collide, stick together and come to rest. Consider to two blocks as a system, the work done on the system by the external forces will be .............. $\mathrm{Joule}$

A body of mass ${m_1}$ moving with uniform velocity of $40 \,m/s$ collides with another mass ${m_2}$ at rest and then the two together begin to move with uniform velocity of $30\, m/s$. The ratio of their masses $\frac{{{m_1}}}{{{m_2}}}$ is

$A$ particle of mass $m$ is released from $a$ height $H$ on $a$ smooth curved surface which ends into a vertical loop of radius $R$, as shown The minimum value of $H$ required so that the particle makes a complete vertical circle is given by