31.
This question has statement $$J$$ and statement II. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - 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 $$f\left( {\frac{1}{2}m{v^2}} \right)$$ then $$f = \left( {\frac{m}{{M + m}}} \right).$$ Statement - II: Maximum energy loss occurs when the particles get stuck together as a result of the collision.
A
Statement - I is true, Statment - II is true, Statement - II is the correct explanation of Statement - I.
B
Statement- I s true, Statment - IT is true, Statement - II is not the correct explanation of Statement - II.
C
Statement - I is true, Statment - II is false.
D
Statement - I is false, Statment - II is true.
Answer :
Statement - I is false, Statment - II is true.
Maximum energy loss $$ = \frac{{{P^2}}}{{2m}} - \frac{{{P^2}}}{{2\left( {m + M} \right)}}\left[ {\because K.E. = \frac{{{P^2}}}{{2m}} = \frac{1}{2}m{v^2}} \right]$$
$$ = \frac{{{P^2}}}{{2m}}\left[ {\frac{M}{{\left( {m + M} \right)}}} \right] = \frac{1}{2}m{v^2}\left\{ {\frac{M}{{m + M}}} \right\}$$
Statement II is a case of perfectly inelastic collision. By comparing the equation given in statement I with above equation, we get
$$f = \left( {\frac{M}{{m + M}}} \right)\,{\text{instead}}\,{\text{of}}\,\left( {\frac{m}{{M + m}}} \right)$$
Hence statement I is wrong and statement II is correct.