A particle of mass $m$ with initial kinetics energy $K$ approaches the origin from $x =+\infty$. Assume that a conservative force acts on it and its potential energy $V ( x )$ is given by $V ( x )=\frac{ K }{\exp \left(3 x / x _0\right)+\exp \left(-3 x / x _0\right)}$ where, $x_0=1 m$. The speed of the particle at $x =0$ is

$A$ $4$ -kilogram disk slides over level ice toward the east at a velocity of $1$ meter per second, as shown. The disk strikes a post and rebounds toward the north at the same speed. The change in the magnitude of the eastward component of the momentum of the disk is ............. $\mathrm{kg}$ $ \cdot $ $\mathrm{m}/ \mathrm{s}$ 

Three balls, $A, B$ and $C$ are released and all reach the point $X$ (shown in the figure). Balls $A$ and $B$ are released from two identical structures, one kept on the ground and the other at height $h$, from the ground as shown in the figure. They take time $t_A$ and $t_B$ respectively to reach $X$ (time starts after they leave the end of the horizontal portion of the structure). The ball $C$ is released from a point at height $h$, vertically above $X$ and reaches $X$ in time $t_C$. Choose the correct option.

A trolley of mass $200\; kg$ moves with a uniform speed of $36\; km / h$ on a frictionless track. A child of mass $20\; kg$ runs on the trolley from one end to the other ( $10\; m$ away) with a speed of $4 \;m s ^{-1}$ relative to the trolley in a direction opposite to the its motion, and Jumps out of the trolley. What is the final speed of the trolley ? How much has the trolley moved from the time the child begins to run?

A bullet weighing $10 \,g$ and moving with a velocity $300 \,m / s$ strikes a $5 \,kg$ block of ice and drop dead. The ice block is kept on smooth surface. The speed of the block after the collision is ........ $cm / s$

A ball is projected from top of a tower with a velocity of $5\,\, m/s$ at an angle of $53^o$ to horizontal. Its speed when it is at a height of $0.45 \,\,m$ from the point of projection is  ........ $m/s$