6.03b Conservation of momentum: 1D two particles

Three particles \(A\), \(B\) and \(C\) lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The masses of \(A\), \(B\) and \(C\) are \(3m\), \(4m\), and \(5m\) respectively. Particle \(A\) is projected with speed \(u\) towards particle \(B\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\).

1. Show that the impulse exerted by \(A\) on \(B\) in this collision has magnitude \(\frac{16}{7}mu\) [7]

After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). After this collision between \(B\) and \(C\), the kinetic energy of \(C\) is \(\frac{72}{245}mu^2\)

1. Find the coefficient of restitution between \(B\) and \(C\). [6]

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 \(\frac{3}{2}u\). [6]

1. 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 smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,

1. show that the speed of \(Q\) immediately after the collision is \(\frac{1}{3}(1 + e)u\), [5]

1. find the range of possible values of \(e\). [4]

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]

1. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

TURN OVER FOR QUESTION 7

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of \(60°\), as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]

1. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]

1. find the distance \(OC\). [6]

END

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of 60°, as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]
2. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]
2. find the distance \(OC\). [6]

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, \(P\), of mass \(2m\), and \(Q\), of mass \(m\), are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between the particles is \(e\), where \(e < 1\). Find, in terms of \(u\) and \(e\),

1. the speed of \(P\) immediately after the collision,
2. the speed of \(Q\) immediately after the collision.

[7]

Three identical particles, \(A\), \(B\) and \(C\), lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The mass of each particle is \(m\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac{2}{3}\).

1. Find, in terms of \(u\),
   1. the speed of \(A\) after this collision,
   2. the speed of \(B\) after this collision.
   [7]
2. Show that the kinetic energy lost in this collision is \(\frac{5}{36}mu^2\) [4]

After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\).

1. Find, in terms of \(u\), the speed of \(C\) immediately after this collision between \(B\) and \(C\). [4]

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_4} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2u\) and the speed of \(Q\) is \(3u\). The coefficient of restitution between \(Q\) and the wall is \(\frac{1}{3}\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).

1. Show that \(v = 2w\). [5]

The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.

1. Find the coefficient of restitution between \(P\) and \(Q\). [8]

\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(OA\) of length 5 m. The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(PQ\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(PQ\), with \(\angle POQ = 120^{\circ}\) and \(OP = OQ = 5\) m, as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt{15}\) m s\(^{-1}\) in a direction perpendicular to the rope \(OA\) and in the plane \(POQ\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m\) kg which is travelling towards her with speed 3 m s\(^{-1}\) and parallel to \(QP\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find

1. the speed of the artiste immediately before she catches the ball, [4]
2. the value of \(m\), [7]
3. the tension in the rope immediately after she catches the ball. [3]

Two smooth spheres \(A\) and \(B\), of masses 60 grams and 90 grams respectively, are at rest on a smooth horizontal table. \(A\) is projected towards \(B\) with speed 4 ms\(^{-1}\) and the particles collide. After the collision, \(A\) and \(B\) move in the same direction as each other, with speeds \(u\) ms\(^{-1}\) and \(6u\) ms\(^{-1}\) respectively. Calculate

1. the value of \(u\), [4 marks]
2. the magnitude of the impulse exerted by \(A\) on \(B\), stating the units of your answer. [3 marks]

\(A\) and \(B\) are now replaced in their original positions and projected towards each other with speeds 2 ms\(^{-1}\) and 8 ms\(^{-1}\) respectively. They collide again, after which \(A\) moves with speed 7 ms\(^{-1}\), its direction of motion being reversed.

1. Find the speed of \(B\) after this collision and state whether its direction of motion has been reversed. [5 marks]

Two railway trucks \(A\) and \(B\), of masses 10 000 kg and 7 000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84 000 Ns. Immediately after the collision, the trucks move together with speed 10 ms\(^{-1}\). Modelling the trucks as particles,

1. find the speed of each truck immediately before the collision. [6 marks]

When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15. Assuming that no other resisting forces act on the trucks, calculate

1. the magnitude of the resisting force on the trucks, [3 marks]
2. the time taken after the collision for the trucks to come to rest. [3 marks]

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]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.

1. Find the distance \(XY\). [2 marks]

A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.

1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]

The particle continues beyond \(Y\) with the same constant velocity.

1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]

When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.

1. Find the velocity vector of the combined particle after the collision. [5 marks]

\(A\), \(B\) and \(C\) are three small spheres of equal radii and masses \(2m\), \(m\) and \(5m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed 6 ms\(^{-1}\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude \(8m\) Ns on \(B\).

1. Find the speed with which \(B\) starts to move. [2 marks]
2. Show that the speed of \(A\) after it collides with \(B\) is 2 ms\(^{-1}\). [3 marks]

After travelling 3 m, \(B\) hits \(C\), which is then travelling towards \(B\) at \(2.2\) ms\(^{-1}\). \(C\) is brought to rest by this impact.

1. Show that the direction of \(B\)'s motion is reversed and find its new speed. [3 marks]
2. Find how far \(B\) now travels before it collides with \(A\) again. [6 marks]
3. State a modelling assumption that you have made about the spheres. [1 mark]