question_answer

A shell at rest at the origin explodes into three fragments of masses \[1\,kg,\,\,2\,kg\] and\[m\,\,kg\]. The \[2\,\,kg\]\[\text{and}\]\[1\,\,kg\] pieces fly off with speeds of \[12\,\,m/s\] along x-axis and \[16\,\,m/s\] along y-axis respectively. If the \[m\,\,kg\] piece flies off with a speed of\[40\,\,m/s\], the total mass of the shell must be

A) \[3.7\,\,kg\]                      

B) \[4\,\,kg\]

C) \[4.5\,\,kg\]                      

D)  \[5\,\,kg\]

Correct Answer: A

Solution :

From principle of conservation of momentum, if no external force acts upon a system of two or more bodies, then the total momentum of the system remains constant. Resolving the momentum components, we have                 \[{{p}_{x}}-{{p}_{m}}\cos \theta =0\]                 \[{{p}_{y}}-{{p}_{m}}\sin \theta =0\] \[\therefore \]  \[{{p}_{m}}=\sqrt{p_{x}^{2}+p_{y}^{2}}\]                 \[m\times 40=\sqrt{{{(2\times 12)}^{2}}+{{(1\times 16)}^{2}}}\] \[\Rightarrow \]               \[m\times 40=29\] \[\Rightarrow \]               \[m=0.7\] Hence, total mass of the shell                 \[M=1+2+0.7=3.7\]