JEE Main & Advanced Physics Work, Energy, Power & Collision / कार्य, ऊर्जा और शक्ति Sample Paper Topic Test - Work Energy Power

  • question_answer
    A body of mass M splits into two parts \[\alpha \] and \[(1-\alpha )\] M by an internal explosion which generates kinetic energy T. After explosion if the two parts move in the same direction as before, their relative speed will be -

    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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