$A$ particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt{x}$,where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = d$ will be:

$A$ demonstration apparatus on a table in the lab is shown in the diagram. It consists of a metal track (shown as a thick solid line in the figure) along which a perfectly spherical marble can roll without slipping. In one run,the marble is released from rest at a height $h$ above the table on the left section,rolls down one side and then up the other side without slipping,briefly stopping when it has reached $h_1$. Assuming the table to be horizontal and neglecting air drag as well as any energy loss due to rolling,which of the following is true?

$A$ force acts on a $3 \,g$ 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 ..... $mJ$.

For a body of mass $1 \ kg$,the $U-x$ graph is shown in the figure. If the body is released from rest at $x = 2 \ m$,then its speed when it crosses $x = 5 \ m$ is:

$A$ car travelling at $80 \ km/h$ can be stopped at a distance of $60 \ m$ by applying brakes. If the same car travels at $160 \ km/h$ and the same braking force is applied,the stopping distance is: (in $m$)

$A$ cricket ball of mass $0.15 \, kg$ is thrown vertically up by a bowling machine so that it rises to a maximum height of $20 \, m$ after leaving the machine. If the part pushing the ball applies a constant force $F$ on the ball and moves a distance of $0.2 \, m$ while launching the ball,the value of $F$ (in $N$) is $(g = 10 \, m/s^2)$.