question_answer

A sphere of mass m, moving with velocity V, enters a hanging bag of sand and stops. If the mass of the bag is M and it is raised by height h, then the velocity of the sphere was                                                              [MP PET 1997]

A)                 \[\frac{M+m}{m}\sqrt{2gh}\]    

B)                 \[\frac{M}{m}\sqrt{2gh}\]

C)                 \[\frac{m}{M+m}\sqrt{2gh}\]

D)                    \[\frac{m}{M}\sqrt{2gh}\]

Correct Answer: A

Solution :

                                    By the conservation of linear momentum Initial momentum of sphere                 = Final momentum of system                 \[mV=(m+M){{v}_{\text{sys}\text{.}}}\]                                               ?(i)                        If the system rises up to height h then by the conservation of energy                 \[\frac{1}{2}(m+M)v_{\text{sys}\text{.}}^{\text{2}}=(m+M)gh\]                              ?(ii)                 Þ \[{{v}_{\text{sys}\text{.}}}=\sqrt{2gh}\]                 Substituting this value in equation (i)                 \[V=\left( \frac{m+M}{m} \right)\ \sqrt{2gh}\]