Multiple successive collisions

A question is this type if and only if it involves three or more particles in a line where collisions occur in sequence, requiring analysis of whether subsequent collisions will occur.

2 questions · Challenging +1.8

Sort by: Default | Easiest first | Hardest first
SPS SPS ASFM 2020 May Q9
10 marks Challenging +1.8
Three particles, \(P\), \(Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(km\) respectively, where \(k\) is a constant. Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly. The coefficient of restitution between each pair of particles is \(e\).
  1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. [9] Given that the mass of \(P\) is \(km\) and that there is a second collision,
  2. write down, in terms of \(u\), \(k\) and \(e\), the speed of \(Q\) after this second collision. [1]
OCR Further Mechanics 2018 September Q8
16 marks Challenging +1.8
A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.
  1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]
\(A\) and \(B\) collide again when \(AO\) is next vertical.
  1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
  2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
  3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]