Question 213: Centre of Mass and Rotational Motion

Question

This question has statement $${\rm I}$$ and statement $${\rm I}$$$${\rm I}$$. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement - $${\rm I}$$: A point particle of mass $$m$$ moving with speed $$\upsilon $$ collides with stationary point particle of mass $$M.$$ If the maximum energy loss possible is given as $$f\left( {{1 \over 2}m{v^2}} \right)$$, then $$f = \left( {{m \over {M + m}}} \right).$$
Statement - $${\rm II}$$: Maximum energy loss occurs when the particles get stuck together as a result of the collision.

Accepted Answer

Initial energy =  $${{{P^2}} \over {2m}}$$, where $$P$$ is the momentum and m is the mass of the moving particle.
Loss of energy is maximum when collision is inelastic means when the particles get stuck together as a result of the collision.
So after collision energy = $${{{P^2}} \over {2\left( {m + M} ight)}}$$
$$	herefore$$ Maximum energy loss $$ = {{{P^2}} \over {2m}} - {{{P^2}} \over {2\left( {m + M} ight)}}$$.
$$\left[  ight.$$ As $$\left. {K.E. = {{{P^2}} \over {2m}} = {1 \over 2}m{v^2}\,\,} ight]$$
$$ = {{{P^2}} \over {2m}}\left[ {{M \over {\left( {m + M} ight)}}} ight] = {1 \over 2}m{v^2}\left\{ {{M \over {m + M}}} ight\}$$
$$	herefore$$ $$f = \left( {{M \over {m + M}}} ight)$$
So statement $$I$$ is wrong.
Statement $${m I}{m I}$$ says "Maximum energy loss occurs when the particles get stuck together as a result of the collision." This is a case of perfectly inelastic collision.
Hence statement $${m I}$$$${m I}$$ is correct.

Answer

Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is true; Statement - $${\rm II}$$ is the correct explanation of Statement - $${\rm I}$$.

Answer

Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is true; Statement - $${\rm II}$$ is not the correct explanation of Statement - $${\rm I}$$.

Answer

Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is false

Answer

Statement - $${\rm I}$$ is false, Statement - $${\rm II}$$ true.

Centre of Mass and Rotational Motion

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