6.03b Conservation of momentum: 1D two particles

6. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of weight 40 N and length 7 m , rests with end \(A\) on rough horizontal ground. The beam rests on a smooth horizontal peg at \(C\), with \(A C = 5 \mathrm {~m}\), as shown in Figure 5.
The beam is inclined at an angle \(\theta\) to the ground, where \(\sin \theta = \frac { 3 } { 5 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg.
The normal reaction between the beam and the peg at \(C\) has magnitude \(P\) newtons.
Using the model,

1. show that \(P = 22.4\)
2. find the magnitude of the resultant force acting on the beam at \(A\).

6. Three particles \(P , Q\) and \(R\) have masses \(3 m , k m\) and 7.5m respectively. The three particles lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 9 }\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 10 u } { 3 ( 3 + k ) }\).
2. Find the range of values of \(k\) for which the direction of motion of \(P\) is reversed as a result of the collision. Following the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\). Given that \(k = 7\) and that \(Q\) is brought to rest by the collision with \(R\),
3. find the total kinetic energy lost in the collision between \(Q\) and \(R\).

7. Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are moving in the same direction along the same straight line on a smooth horizontal surface, with \(B\) in front of \(A\). Particle \(A\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). The direction of motion of both particles is not changed by the collision. Immediately after the collision, \(A\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. 1. Show that \(w = \frac { 23 } { 9 }\).
   2. Find the value of \(v\). When \(A\) and \(B\) collide they are 3 m from a smooth vertical wall which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) hits the wall and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). There is a second collision between \(A\) and \(B\) at a point \(d \mathrm {~m}\) from the wall.
2. Find the value of \(d\).

7. Three particles \(A , B\) and \(C\) lie at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The particles \(A\), \(B\) and \(C\) have mass \(6 m\), 4 \(m\) and \(m\) respectively. Particle \(A\) is projected towards \(B\) with speed \(3 u\) and \(A\) collides directly with \(B\). Immediately after this collision, the speed of \(B\) is \(w\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 6 }\).

1. Show that \(w = \frac { 21 } { 10 } u\).
2. Express the total kinetic energy of \(A\) and \(B\) lost in the collision as a fraction of the total kinetic energy of \(A\) and \(B\) immediately before the collision. After being struck by \(A\), the particle \(B\) collides directly with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). After the collision between \(B\) and \(C\), there are no further collisions between the particles.
3. Find the range of possible values of \(e\).
   

7. Three particles \(A\), \(B\) and \(C\) have masses \(2 m , 3 m\) and \(4 m\) respectively. The particles lie at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). The kinetic energy of \(A\) immediately after the collision is one ninth of the kinetic energy of \(A\) immediately before the collision. Given that the direction of motion of \(A\) is unchanged by the collision,

1. find the value of \(e\). After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). The coefficient of restitution between \(B\) and \(C\) is \(f\), where \(f < \frac { 3 } { 4 }\). The speed of \(B\) immediately after the collision with \(C\) is \(V\).
2. 1. Express \(V\) in terms of \(f\) and \(u\).
   2. Hence show that there will be a second collision between \(A\) and \(B\).
      

7. Particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal surface. Particle \(A\) collides directly with particle \(B\) of mass \(m\), which is moving along the same straight line and in the same direction as \(A\). Immediately before the collision, the speed of \(B\) is \(u\).
As a result of the collision, the direction of motion of \(B\) is unchanged and the kinetic energy gained by \(B\) is \(\frac { 48 } { 25 } m u ^ { 2 }\)

1. Find the coefficient of restitution between \(A\) and \(B\).
   (8) After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(f\). Given that the speed of \(B\) immediately after first hitting the wall is equal to the speed of \(A\) immediately after its first collision with \(B\),
2. find the value of \(f\).

1. Particles \(A , B\) and \(C\), of masses \(2 m , m\) and \(3 m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(2 u\) and collides directly with \(B\).

The coefficient of restitution between each pair of particles is \(e\).

1. 1. Show that the speed of \(B\) immediately after the collision with \(A\) is \(\frac { 4 } { 3 } u ( 1 + e )\)
   2. Find the speed of \(A\) immediately after the collision with \(B\). At the instant when \(A\) collides with \(B\), particle \(C\) is projected with speed \(u\) towards \(B\) so that \(B\) and \(C\) collide directly.
2. Show that there will be a second collision between \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-27_2644_1840_118_111}

6. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface so that the particles collide directly.
The mass of \(P\) is \(k m\) and the mass of \(Q\) is \(m\).
Immediately before the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\). Immediately after the collision, \(P\) and \(Q\) are moving in the same direction, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(2 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\) The magnitude of the impulse received by \(Q\) in the collision is \(5 m v\)

1. Find (i) \(y\) in terms of \(v\) (ii) \(x\) in terms of \(v\) (iii) the value of \(k\)
2. Find, in terms of \(m\) and \(v\), the total kinetic energy lost in the collision between \(P\) and \(Q\).

1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(5 m\) are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(k u\).
Immediately after the collision, the speed of \(P\) is \(2 v\) and the speed of \(Q\) is \(v\).
The direction of motion of each particle is reversed by the collision.
In the collision, \(P\) receives an impulse of magnitude \(15 m v\).

1. Show that \(u = 3 v\).
2. Find the value of \(k\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
3. Find the value of \(e\). The total kinetic energy lost in the collision is \(\lambda m v ^ { 2 }\)
4. Find the value of \(\lambda\).

1. A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(2 m\) are at rest on a smooth horizontal plane.

Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide. The coefficient of restitution between the particles is \(e\). As a result of the collision, the direction of motion of \(P\) is reversed.

1. Find, in terms of \(u\) and \(e\), the speed of \(P\) after the collision. After the collision, \(Q\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\) Given that there is a second collision between \(P\) and \(Q\)
2. find the full range of possible values of \(e\).

8. Particles \(A , B\) and \(C\), of masses \(4 m , k m\) and \(2 m\) respectively, lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(3 u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\) Find

1. the speed of \(A\) immediately after the collision with \(B\), giving your answer in terms of \(u\) and \(k\),
2. the range of values of \(k\) for which \(A\) and \(B\) will both be moving in the same direction immediately after they collide. After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\). Given that \(k = 4\),
3. show that there will not be a second collision between \(A\) and \(B\).
   DO NOT WRITEIN THIS AREA

8. A particle \(A\) of mass \(3 m\) lies at rest on a smooth horizontal floor. A particle \(B\) of mass \(2 m\) is moving in a straight line on the floor with speed \(u\) when it collides directly with \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). As a result of the collision the direction of motion of \(B\) is reversed.

1. Find an expression, in terms of \(u\) and \(e\), for
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. The particle \(A\) subsequently strikes a smooth vertical wall. The wall is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(A\) and the wall is \(\frac { 1 } { 7 }\) There is a second collision between \(A\) and \(B\).
2. Show that \(\frac { 2 } { 3 } < e < \frac { 16 } { 19 }\)

7. A particle, \(P\), of mass \(k m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal surface. Particle \(P\) collides directly with another particle, \(Q\), of mass \(2 m\) which is moving with speed \(u\) in the same direction along the same straight line. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Given that immediately after the collision \(P\) and \(Q\) are moving in opposite directions and the speed of \(Q\) is \(\frac { 3 } { 2 } u\),

1. find the range of possible values of \(e\). It is now also given that \(e = \frac { 7 } { 8 }\).
2. Show that the kinetic energy lost by \(P\) in the collision with \(Q\) is \(\frac { 11 } { 8 } m u ^ { 2 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision, \(Q\) hits a fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) rebounds from the wall and moves so that \(P\) and \(Q\) collide directly at the point \(B\).
3. Find, in terms of \(d\) and \(u\), the time interval between the collision at \(A\) and the collision at \(B\).
   
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6. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\).
Immediately after the collision, \(A\) and \(B\) are moving in opposite directions with the same speed \(v\).
In the collision, \(A\) receives an impulse of magnitude \(5 m v\).

1. Find the coefficient of restitution between \(A\) and \(B\).
   (6) After the collision with \(A\), particle \(B\) strikes a smooth fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of the particles.
   The coefficient of restitution between \(B\) and the wall is \(f\).
   As a result of its collision with \(A\) and with the wall, the total kinetic energy lost by \(B\) is \(E\). As a result of its collision with \(B\), the kinetic energy lost by \(A\) is \(2 E\).
2. Find the value of \(f\). \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-19_2664_107_106_6}
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7. Particle \(A\) has mass \(m\) and particle \(B\) has mass \(2 m\). The particles are moving in the same direction along the same straight line on a smooth horizontal surface.
Particle \(A\) collides directly with particle \(B\).
Immediately before the collision, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 + 2 e } { 3 } u\)
   2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\).
      The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\) Particle \(B\) rebounds and there is a second collision between \(A\) and \(B\).
      The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall.
      The time between the two collisions is \(T\).
      Given that \(e = \frac { 1 } { 2 }\)
2. find \(T\) in terms of \(d\) and \(u\).

1. Particle \(P\) has mass \(4 m\) and particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of each particle is reversed as a result of the collision.
The total kinetic energy of \(P\) and \(Q\) after the collision is half of the total kinetic energy of \(P\) and \(Q\) before the collision.

1. Show that \(y = \frac { 8 } { 3 } u\) The coefficient of restitution between \(P\) and \(Q\) is \(e\).
2. Find the value of \(e\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is no second collision between \(P\) and \(Q\),
3. find the range of possible values of \(f\). Given that \(f = \frac { 1 } { 4 }\)
4. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) as a result of its impact with the wall.

2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).

1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
2. find the magnitude of the impulse exerted on \(Q\) in the collision.
   

8. A small ball A of mass 3 m 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. A fter 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 \(\frac { 2 } { 5 }\).
2. 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.
3. Show that \(T = \frac { 112 a } { 15 u }\).

6. A smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another smooth sphere \(B\) of mass \(3 m\), which is at rest on the table. The coefficient of restitution between \(A\) and \(B\) is \(e\). The spheres have the same radius and are modelled as particles.

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } ( 1 + e ) u\).
2. Find the speed of \(A\) immediately after the collision. Immediately after the collision the total kinetic energy of the spheres is \(\frac { 1 } { 6 } m u ^ { 2 }\).
3. Find the value of \(e\).
4. Hence show that \(A\) is at rest after the collision.

6. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2 m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) after the collision is \(\frac { 1 } { 5 } u ( 9 e + 4 )\). As a result of the collision, the direction of motion of \(P\) is reversed.
2. Find the range of possible values of \(e\). Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\frac { 32 } { 5 } m u\),
3. find the value of \(e\).
   (4)

2. A particle \(A\) of mass \(4 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3 m\) moving with speed \(2 u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4 e u\).

1. Show that \(e = \frac { 3 } { 4 }\).
2. Find the total kinetic energy lost in the collision.

1. Particles \(A , B\) and \(C\) of masses \(4 m , 3 m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, and collide directly.

As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(k v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).

1. Show that \(u = 3 v\).
2. Find the value of \(k\). Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) so that \(B\) and \(C\) collide directly.
3. Show that there is no further collision between \(A\) and \(B\).

2. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4 m\) moving on the plane with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the speed of \(Q\) immediately after the collision. Given that the direction of motion of \(P\) is reversed by the collision,
2. find the range of possible values of \(e\).