A) momentum is \[\frac{m\upsilon M}{M+m}\]
B) kinetic energy is \[\frac{m{{V}^{2}}}{2}\]
C) momentum is \[\frac{m\upsilon (M+m)}{M}\]
D) kinetic energy is \[\frac{{{m}^{2}}{{v}^{2}}}{2(M+m)}\]
Correct Answer: D
Solution :
The combined velocity of bag + bullet system, \[v'=m\frac{v}{m+M}\] Now, we observe that the options talk about momentum and kinetic energy, so we will calculate the value of these quantities one by one: (i) Momentum\[=(m+M)v'=mv\] Note that, the final momentum must equal the initial momentum, so this value of final momentum could have been directly written. (ii) Kinetic energy\[=\frac{1}{2}(m+M)v{{'}^{2}}\] \[=\frac{1}{2}(m+M){{\left[ m\frac{v}{m+M} \right]}^{2}}\] \[=\frac{1}{2}(m+M)\frac{{{m}^{2}}{{v}^{2}}}{{{(m+M)}^{2}}}\] \[={{m}^{2}}\frac{{{v}^{2}}}{2(M+m)}\]
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