A bullet when fired at a target with a velocity of $100\,\,m/sec$ penetrates one metre into it. If the bullet is fired at a similar target with a thickness $0.5\,\,metre,$  then it will emerge from it with a velocity of

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

Assertion $A$ : Body $'P'$ having mass $M$ moving with speed $'u'$ has head-on collision elastically with another body $'Q'$ having mass $'m'$ initially at rest. If $m< < M,$ body $'Q'$ will have a maximum speed equal to $'2u'$ after collision.

Reason $R$ : During elastic collision, the momentum and kinetic energy are both conserved.

In the light of the above statements, choose the most appropriate answer from the options given below:

At high altitude, a body explodes at rest into two equal fragments with one fragment receiving horizontal velocity of $10 \,m/s$. Time taken by the two radius vectors connecting point of explosion to fragments to make $90^o $ is ............ $\mathrm{s}$

Find work done by friction if block reaches to the end with constant velocity

A small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $L = L _0$ the particle speed is $v = v _0$. The piston is moved inward at a very low speed $V$ such that $V \ll \frac{ dL }{ L } v _0$, where $dL$ is the infinitly small displacement of the piston. Which of the following statement($s$) is/are correct?

$(1)$ The rate at which the particle strikes the piston is $v / L$

$(2)$ After each collision with the piston, the particle speed increases by $2 V$

$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L _0$ to $\frac{1}{2} L _0$

$(4)$ If the piston moves inward by $d L$, the particle speed increases by $2 v \frac{d L}{L}$

A body of mass $2\,kg$ makes an elastic collision with a second body at rest and continues to move in the original direction but with one fourth of its original speed. What is the mass of the second body? ................ $ \mathrm{kg}$