(D) Initial momentum of the system = $mv$.Final momentum of the system = $(m + M)V$,where $V$ is the common velocity of the system after collision.By

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Kinetic energy is $\frac{m^2v^2}{2(M + m)}$

(D) Initial momentum of the system = $mv$.Final momentum of the system = $(m + M)V$,where $V$ is the common velocity of the system after collision.By the law of conservation of linear momentum,$mv = (m + M)V$.Therefore,the common velocity $V = \frac{mv}{M + m}$.The kinetic energy of the combined system is $K = \frac{1}{2}(m + M)V^2$.Substituting the value of $V$,we get $K = \frac{1}{2}(m + M) \left( \frac{mv}{M + m} \right)^2$.$K = \frac{1}{2}(m + M) \frac{m^2v^2}{(M + m)^2} = \frac{m^2v^2}{2(M + m)}$.Thus,option $D$ is correct.

Class 11 Physics Work, Energy, Power and Collision Perfectly Inelastic Collision

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$A$ bag (mass $M$) hangs by a long thread and a bullet (mass $m$) comes horizontally with velocity $v$ and gets caught in the bag. Then for the combined (bag $+$ bullet) system

A
Momentum is $\frac{mvM}{M + m}$
B
Kinetic energy is $\frac{mv^2}{2}$
C
Momentum is $\frac{mv(M + m)}{M}$
D
Kinetic energy is $\frac{m^2v^2}{2(M + m)}$

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Work, Energy, Power and Collision

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Perfectly Inelastic Collision

$A$ body of mass $m$ moving with velocity $v$ collides head-on with another body of mass $2m$ which is initially at rest. If the collision is perfectly inelastic,what is the loss in kinetic energy of the system?

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$A$ body of mass $m$ moves with velocity $v$ in the East direction,and another body of the same mass moves with velocity $v$ in the North direction. After they collide and stick together,what is the combined velocity of the resulting body of mass $2m$?

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$A$ body of mass $2 \, kg$ moving with a velocity $(\hat{i} + \hat{j} + \hat{k}) \, m/s$ collides with another body of mass $5 \, kg$ moving with velocity $(\hat{i} - 2\hat{j} + 3\hat{k}) \, m/s$. If they stick together,the velocity (in $m/s$) of the composite body is:

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$A$ block of mass $0.50 \ kg$ is moving with a speed of $2.00 \ m/s$ on a smooth surface. It strikes another mass of $1.00 \ kg$ at rest and then they move together as a single body. The energy loss during the collision is ............. $J$.

MediumAIEEE 2008

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$A$ body of mass $m$ moving with a constant velocity $v$ hits another body of the same mass moving with the same velocity $v$ but in the opposite direction and sticks to it. The velocity of the compound body after collision is

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$A$ bag (mass $M$) hangs by a long thread and a bullet (mass $m$) comes horizontally with velocity $v$ and gets caught in the bag. Then for the combined (bag $+$ bullet) system

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$A$ body of mass $m$ moving with velocity $v$ collides head-on with another body of mass $2m$ which is initially at rest. If the collision is perfectly inelastic,what is the loss in kinetic energy of the system?

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