In two different experiments, an object of mass $5\; kg$ moving with a speed of $25\; ms ^{-1}$ hits two different walls and comes to rest within

$(i)$ $3$ second, $(ii)$ $5$ seconds, respectively.

Choose the correct option out of the following :

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 $e ^{-1}=0.37$

When the speed of a moving body is doubled

A girl riding a bicycle along a straight road with a speed of $5\,ms^{-1}$ throws a stone of mass $0.5\, kg$ which has a speed of $15\,ms^{-1}$ with respect to the ground along her direction of motion. The mass of the girl and bicycle is $ 50\, kg$. Does the speed of the bicycle change after the stone is thrown ? What is the change in speed, if so ?

The force $F$ acting on a particle of mass $m$ is indicated by the force-time graph shown below. The change in momentum of the particle over the time interval from zero to $8\, s$ is   ………. $N-s$