AS June 2024 Q4

EdexcelAS paperCurrent spec11 marksCollisions: 1 Sphere DirectCollisions: 2 Spheres Direct

4.

Figure 2: a vertical wall on the left of a horizontal plane, with particle P of mass m and particle Q of mass 4m at rest on the plane, P nearer the wall
Figure 2

A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4m\) are at rest on a smooth horizontal plane, as shown in Figure 2.

Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.

The coefficient of restitution between the particles is \(e\), where \(e \gt \dfrac{1}{4}\)

As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\dfrac{u}{5}(4e-1)\).

(a) Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. (3)

After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\).

The coefficient of restitution between \(P\) and the wall is \(f\), where \(f \gt 0\)

Given that \(e = \dfrac{3}{4}\)

(b) find, in terms of \(m\), \(u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. (4)

After its impact with the wall, \(P\) goes on to collide with \(Q\) again.

(c) Find the complete range of possible values of \(f\). (4)
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