6.03j Perfectly elastic/inelastic: collisions

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]

Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision. [3]

Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan\theta = \frac{3}{4}\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\). Immediately after \(B\) collides with the wall, the kinetic energy of \(B\) is \(\frac{5}{27}\) of the kinetic energy of \(B\).

1. Find the possible values of \(e\). [7]

A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Show that the speed of \(B\) after the collision is \(\frac{5}{3}u\). [6]
2. Find the speed of \(A\) after the collision. [2]

Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,

1. find the range of possible values for \(e\). [8]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(u\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]

In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{3}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]

A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.

1. Find the speeds of \(A\) and \(B\) immediately after the collision. [7]
2. Find the range of possible values of \(e\). [2]

After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).

1. Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\). [6]

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]

\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).

1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]

For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,

1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
2. find the ratio \(d_1 : d_2\). [5]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is 2.5 m s\(^{-1}\) and the speed of \(B\) is 1.3 m s\(^{-1}\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{12}{5}\) as shown in Fig. 1.

1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. [4]

The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Find, to one decimal place, the speed of each sphere after the collision. [9]

\includegraphics{figure_3} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg. After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m, and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m. The angle \(AXB\) is 90°. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van. [5]

The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(AX\) and the car to a constant horizontal force of 300 N along \(BX\).

1. Find the speed of the car immediately before the collision. [9]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] The vector \(\mathbf{n} = (-\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j})\) and the vector \(\mathbf{p} = (-\frac{4}{5}\mathbf{i} + \frac{3}{5}\mathbf{j})\) are perpendicular unit vectors.

1. Verify that \(\frac{3}{5}\mathbf{n} + \frac{4}{5}\mathbf{p} = (\mathbf{i} + 3\mathbf{j})\). [2]

A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed vertical wall which is parallel to \(\mathbf{p}\). Immediately after the collision the velocity of \(S\) is \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between \(S\) and the wall is \(\frac{3}{5}\).

1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the velocity of \(S\) immediately before the collision. [5]
2. Find the energy lost in the collision. [3]

A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\sqrt{3}\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt{\left(\frac{37ga}{5}\right)}\).

1. Show that \(S\) strikes the wall with speed \(\sqrt{\left(\frac{27ga}{5}\right)}\). [4] Given that the loss in kinetic energy due to the impact with the wall is \(\frac{3mga}{5}\),
2. find the coefficient of restitution between \(S\) and the wall. [7]

Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60°\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{3u}{4(k + 1)}\). [6] Immediately after the collision the direction of motion of \(A\) makes an angle arctan \((2\sqrt{3})\) with the direction of motion of \(B\).
2. Show that \(k = \frac{1}{2}\). [6]
3. Find the loss of kinetic energy due to the collision. [4]

A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find

1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
2. the magnitude of the impulse received by the ball as it strikes the plane. [2]

\includegraphics{figure_1} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]

A uniform rod \(AB\) of mass \(4m\) is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), through \(A\). The rod is hanging vertically at rest when it is struck at its end \(B\) by a particle of mass \(m\). The particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to \(L\), and after striking the rod it rebounds in the opposite direction with speed \(v\). The coefficient of restitution between the particle and the rod is \(1\). Show that \(u = 7v\). [7]

\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.

1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
   1. the total kinetic energy of the spheres before the collision, [1]
   2. the total kinetic energy of the spheres after the collision. [2]
3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
4. Comment on the cases when
   1. \(\lambda = 1\),
   2. \(\lambda = \frac{25}{52}\). [3]

A small ball of mass \(m\) kg is held at a height of \(78.4\) m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is \(e\). The ball continues to bounce until it comes to rest after \(6\) seconds.

1. Determine the value of \(e\). [8]
2. Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is \(23.52\) Ns, determine the value of \(m\). [2]

\includegraphics{figure_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.

1. Find the value of \(e\). [4]
2. Find the size of angle \(\beta\). [4]