A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m =0.4 kg$ is at rest on this surface. An impulse of $1.0 N s$ is applied to the block at time $t =0$ so that it starts moving along the $x$-axis with a velocity $v ( t )= v _0^{ e ^{-1 / \tau}}$, where $v_0$ is a constant and $\tau=4 s$. The displacement of the block, in metres, at $t=\tau$ is. . . . . . . Take $e ^{-1}=0.37$

This question has Statement $1$ and Statement $2$. Of the four choices given after the Statements, choose the one that best describes the two Statements.

Statement $1$: If you push on a cart being pulled by a horse so that it does not move, the cart pushes you back with an equal and opposite force.

Statemf'nt $2$: The cart does not move because the force described in statement $1$ cancel each other

Two trains $A$ and $B$ are running in the same direction on parallel rails such that $A$ is faster than $B$ . Packets of equal weight are transferred between them. What will happen due to this

A rigid ball of mass $m$ strikes a rigid wall at $60^o $ and gets reflected without loss of speed as shown in the figure. The value of impulse imparted by the wall on the ball will be

The linear momentum $p$ of a body moving in one dimension varies with time according to the equation $p = a + b{t^2}$ where a and b are positive constants. The net force acting on the body is

A women throws an object of mass $500\,g$ with a speed of $25\, ms^{-1}$.

$(a)$ What is the impulse imparted to the object ?

$(b)$ If the object hits a wall and rebounds with half the original speed, what is the change in momentum of the object ?