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Question

A particle of mass m collided with a stationary particle of mass M deviated by an angle π2 whereas the particle M recoiled at an angle θ=30 to the direction of the initial motion of the particle m. How much (in per cent) has the kinetic energy of this system changed after the collision, if Mm=5.0?

Solution
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Since, no external impulsive force is effective on the system "M+m", its total momentum along any direction will remain conserved.
So from px= const.,

mu=Mv1cosθ or, v1=mMucosθ (1)

and from py= const.,

mv2=Mv1sinθ or, v2=Mmv1sinθ=utanθ, [using (1)]

Final kinetic energy of the system

Tf=12mv22+12Mv21

And initial kinetic energy of the system =12mu2

So, % change =TfTiTi×100

=12mu2tan2θ+12Mm2M2u2cos2θ12mu212mu2×100

=12u2tan2θ+12mMu2sec2θ12u212u2×100

=(tan2θ+mMsec2θ1)×100

and putting the values of θ and mM, we get % of change in kinetic energy =40%

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