4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively.
Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below.
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The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .
- State how you can tell from the information given above that kinetic energy is lost when A collides with B .
- Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision.
Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
- Given that a third collision occurs, determine the range of possible values for \(u\).
- State one limitation of the model used in this question.