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Theoretical Verification of Law of Conservation

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Question

Find the increment of the kinetic energy of the closed system comprising two spheres of masses $m_{1}$ and $m_{2}$ due to their perfectly inelastic collision, if the initial velocities of the spheres were equal to $_{1}$ and $_{2}$ ?

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Solution

For perfectly inelastic collision, in the centre of mass(C.M) frame, final kinetic energy of the colliding system (both spheres) becomes zero. Hence initial kinetic energy of the system in C.M. frame completely turns into the internal energy ( $Q$ ) of the formed body. Hence
$Q =_{i} = \frac{1}{2} μ {|_{1} -_{2} |}^{2}$
Now from energy conservation $Δ T = - Q = - \frac{1}{2} μ {|_{1} -_{2} |}^{2}$ ,
In lab frame the same result is obtained as $Δ T = \frac{1}{2} \frac{{(m_{1}_{1} + m_{2}_{2})}^{2}}{m_{1} + m_{2}} - \frac{1}{2} m_{1} {|_{1} |}^{2} + m_{2} {|_{2} |}^{2}$
$= - \frac{1}{2} μ {|_{1} -_{2} |}^{2}$

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