JEE MAIN - Physics (2013 (Offline) - No. 26)
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.
Statement - $${\rm II}$$: Maximum energy loss occurs when the particles get stuck together as a result of the collision.
Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is true; Statement - $${\rm II}$$ is the correct explanation of Statement - $${\rm I}$$.
Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is true; Statement - $${\rm II}$$ is not the correct explanation of Statement - $${\rm I}$$.
Statement - $${\rm I}$$ is true, Statement - $${\rm II}$$ is false
Statement - $${\rm I}$$ is false, Statement - $${\rm II}$$ true.
Explanation
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 = $${{{P^2}} \over {2\left( {m + M} \right)}}$$
$$\therefore$$ Maximum energy loss $$ = {{{P^2}} \over {2m}} - {{{P^2}} \over {2\left( {m + M} \right)}}$$.
$$\left[ \right.$$ As $$\left. {K.E. = {{{P^2}} \over {2m}} = {1 \over 2}m{v^2}\,\,} \right]$$
$$ = {{{P^2}} \over {2m}}\left[ {{M \over {\left( {m + M} \right)}}} \right] = {1 \over 2}m{v^2}\left\{ {{M \over {m + M}}} \right\}$$
$$\therefore$$ $$f = \left( {{M \over {m + M}}} \right)$$
So statement $$I$$ is wrong.
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 = $${{{P^2}} \over {2\left( {m + M} \right)}}$$
$$\therefore$$ Maximum energy loss $$ = {{{P^2}} \over {2m}} - {{{P^2}} \over {2\left( {m + M} \right)}}$$.
$$\left[ \right.$$ As $$\left. {K.E. = {{{P^2}} \over {2m}} = {1 \over 2}m{v^2}\,\,} \right]$$
$$ = {{{P^2}} \over {2m}}\left[ {{M \over {\left( {m + M} \right)}}} \right] = {1 \over 2}m{v^2}\left\{ {{M \over {m + M}}} \right\}$$
$$\therefore$$ $$f = \left( {{M \over {m + M}}} \right)$$
So statement $$I$$ is wrong.
Statement $${\rm I}{\rm I}$$ 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 $${\rm I}$$$${\rm I}$$ is correct.
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