6.03i Coefficient of restitution: e

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5m\) and \(2m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{1}{7}u(1 + 15e)\) and find an expression for the speed of \(A\). [4]

In the collision, the speed of \(A\) is halved and its direction of motion is reversed.

1. Find the value of \(e\). [2]
2. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). [3]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
2. Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]

Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).

1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]

Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.

1. Find the set of possible values of \(e\). [6]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

One end of a light elastic string of natural length \(0.6 \text{ m}\) and modulus of elasticity \(24 \text{ N}\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.4 \text{ kg}\) which hangs in equilibrium vertically below \(O\).

1. Calculate the extension of the string. [2]

\(P\) is projected vertically downwards from the equilibrium position with speed \(5 \text{ m s}^{-1}\).

1. Calculate the distance \(P\) travels before it is first at instantaneous rest. [4]

When \(P\) is first at instantaneous rest a stationary particle of mass \(0.4 \text{ kg}\) becomes attached to \(P\).

1. Find the greatest speed of the combined particle in the subsequent motion. [4]

\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere \(A\), with mass \(m\), is at rest on a smooth horizontal surface and is at a distance \(d\) from each wall. An identical smooth sphere \(B\) is moving on the horizontal surface with speed \(u\) at an angle \(\theta\) with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the value of \(\tan \theta\). [4]
2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]

\includegraphics{figure_3} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2u\) and \(3u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres and \(B\)'s direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{3}\).

1. Find the value of \(\tan \theta\). [3]
2. Find the total loss of kinetic energy as a result of the collision. [2]
3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. [2]

A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{2}{9}u\) and \(\frac{8}{9}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]
2. Explain why there must be a third collision between \(P\) and \(Q\). [1]

A particle \(A\) of mass \(2m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac{8}{3}u\) and the direction of motion of \(B\) is reversed.

1. Calculate the coefficient of restitution between \(A\) and \(B\). [6]
2. Show that the kinetic energy lost in the collision is \(7mu^2\). [3]

After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac{14}{3}mu\).

1. Calculate the coefficient of restitution between \(B\) and the wall. [4]

A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal table. Another particle \(Q\) of mass \(km\) is at rest on the table. The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. After the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(3v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{2}\).

1. Find, in terms of \(v\) only, the speed of \(P\) before the collision. [3]
2. Find the value of \(k\). [3]

After being struck by \(P\), the particle \(Q\) collides directly with a particle \(R\) of mass \(11m\) which is at rest on the table. After this second collision, \(Q\) and \(R\) have the same speed and are moving in opposite directions. Show that

1. the coefficient of restitution between \(Q\) and \(R\) is \(\frac{1}{4}\), [4]
2. there will be a further collision between \(P\) and \(Q\). [2]

A particle \(P\) of mass \(2m\) is moving with speed \(2u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac{3}{2}u\). [5]
2. Find the total kinetic energy lost in the collision. [5]

After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).

1. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\). [7]

Two particles \(A\) and \(B\) move on a smooth horizontal table. The mass of \(A\) is \(m\), and the mass of \(B\) is \(4m\). Initially \(A\) is moving with speed \(u\) when it collides directly with \(B\), which is at rest on the table. As a result of the collision, the direction of motion of \(A\) is reversed. The coefficient of restitution between the particles is \(e\).

1. Find expressions for the speed of \(A\) and the speed of \(B\) immediately after the collision. [7]

In the subsequent motion, \(B\) strikes a smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{4}{5}\). Given that there is a second collision between \(A\) and \(B\),

1. show that \(\frac{1}{4} < e < \frac{9}{16}\). [5]

Given that \(e = \frac{1}{2}\),

1. find the total kinetic energy lost in the first collision between \(A\) and \(B\). [3]

A small ball \(A\) of mass \(3m\) is moving with speed \(u\) in a straight line on a smooth horizontal table. The ball collides directly with another small ball \(B\) of mass \(m\) moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\). The balls have the same radius and can be modelled as particles.

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision.
   (7)

After the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\).

1. Find the speed of \(B\) immediately after hitting the wall. (2)

The first collision between \(A\) and \(B\) occurred at a distance \(4a\) from the wall. The balls collide again \(T\) seconds after the first collision.

1. Show that \(T = \frac{112a}{15u}\). (6)

A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision. Show that \(e > \frac{1}{3}\). [8]

Two particles \(P\) and \(Q\), of masses \(2m\) and \(m\) respectively, are on a smooth horizontal table. Particle \(Q\) is at rest and particle \(P\) collides directly with it when moving with speed \(u\). After the collision the total kinetic energy of the two particles is \(\frac{3}{4}mu^2\). Find

1. the speed of \(Q\) immediately after the collision, [10]
2. the coefficient of restitution between the particles. [3]

Two smooth spheres \(X\) and \(Y\), of masses \(x\) kg and \(y\) kg respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6\) ms\(^{-1}\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2\) ms\(^{-1}\) and \(Y\) moves with speed \(3\) ms\(^{-1}\).

1. Calculate the two possible values of the ratio \(x : y\). \hfill [6 marks]
2. State a modelling assumption that you have made concerning \(X\) and \(Y\). \hfill [1 mark]

\(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k\) ms\(^{-1}\), colliding again with \(X\) which is still moving at \(2\) ms\(^{-1}\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,

1. show that \(k > 1.5\). \hfill [4 marks]

Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find

1. the speed of \(Q\) before the collision, [4 marks]
2. the speed of \(P\) after the collision, [4 marks]
3. the kinetic energy, in J, lost in the impact. [4 marks]

Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
2. Show that \(e > \frac{1}{9}\). [2 marks]

\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).

1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]

Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]

Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]

Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).

1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]

Two railway trucks \(A\) and \(B\), whose masses are \(6m\) and \(5m\) respectively, are moving in the same direction along a straight track with speeds \(5u\) and \(3u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(kv\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\). Modelling the trucks as particles,

1. show that
   1. \(v = \frac{45u}{5k + 6}\),
   2. \(v = \frac{2eu}{k - 1}\).
   [8 marks]
2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). [5 marks]

A heavy ball, of mass 2 kg, rolls along a horizontal surface. It strikes a vertical wall at a speed of 4 ms\(^{-1}\) and rebounds. The coefficient of restitution between the ball and the wall is 0.4. Find the kinetic energy lost in the impact. [5 marks]

Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 ms\(^{-1}\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.

1. Show that the speed of \(B\) after the impact cannot be more than 3 ms\(^{-1}\). [5 marks]

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(e < \frac{3}{10}\). [5 marks]
2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]

\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).

1. Find the coefficient of restitution between \(A\) and \(B\). [4]
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]

Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.

1. Show that there will be another collision. [3]

A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find

1. the speed of \(A\) immediately after the collision, [4]
2. the distance \(A\) moves up the plane after the collision. [2]

The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.

1. Find the value of \(m\). [3]