A) \[\sqrt{\frac{T}{(1-\alpha )M}}\]
B) \[\sqrt{\frac{2T}{\alpha (1-\alpha )M}}\]
C) \[\sqrt{\frac{T}{2(1-\alpha )M}}\]
D) \[\sqrt{\frac{2T}{(1-a)M}}\]
Correct Answer: B
Solution :
| [b] Let the speed of the body before explosion be u. After explosion, if the two parts move with velocities \[{{u}_{1}}\] and \[{{u}_{2}}\] in the same direction, then according to conservation of momentum, |
| Or \[M{{u}_{1}}+\left( 1-\alpha \right)M\,{{u}^{2}}=Mu\] |
| The kinetic energy T liberated during explosion is given |
| by \[T=\frac{1}{2}\alpha Mu_{1}^{2}+\frac{1}{2}\left( 1-\alpha \right)Mu_{2}^{2}-\frac{1}{2}M{{u}^{2}}\] |
| \[=\frac{1}{2}\alpha M\,u_{1}^{2}+\frac{1}{2}\left( 1-\alpha \right)M\,u_{2}^{2}-\frac{1}{2M}\] |
| \[{{\left[ \alpha M{{u}_{1}}+\left( 1-\alpha \right)M{{u}_{2}} \right]}^{2}}\] |
| \[=\frac{1}{2}M\alpha \left( 1-\alpha \right)\left[ u_{1}^{2}+u_{2}^{2}-2{{u}_{1}}{{u}_{2}} \right]\] |
| \[{{\left( {{u}_{1}}-{{u}_{2}} \right)}^{2}}=\frac{2T}{\alpha \left( 1-\alpha \right)M}\] |
| \[\Rightarrow \left( {{u}_{1}}-{{u}_{2}} \right)=\sqrt{\frac{2T}{\alpha \left( 1-\alpha \right)M}}\] |
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