Collision followed by wall impact

5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(3 m\) respectively, are moving in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly and, as a result of the collision, the direction of motion of \(P\) is reversed and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(\frac { 3 v } { 2 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\).

1. Find
   1. the speed of \(P\) immediately before the collision,
   2. the speed of \(Q\) immediately before the collision. After the collision with \(P\), the particle \(Q\) moves on the plane and strikes at right angles a fixed smooth vertical wall and rebounds. The coefficient of restitution between \(Q\) and the wall is \(e\). Given that there is a further collision between the particles,
2. find the range of possible values of \(e\).

8. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately after the collision, \(A\) and \(B\) are moving in the same direction with speeds \(\frac { u } { 3 }\) and \(u\) respectively. In the collision, \(A\) receives an impulse of magnitude 8mu.

1. Find the coefficient of restitution between \(A\) and \(B\). When \(A\) and \(B\) collide they are at a distance \(d\) from a smooth vertical wall, which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) collides directly with the wall and rebounds so that there is a second collision between \(A\) and \(B\). This second collision takes place at distance \(x\) from the wall. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 4 }\)
2. find \(x\) in terms of \(d\).
   

   END

4. Two small balls, \(A\) and \(B\), are moving in opposite directions along the same straight line on smooth horizontal ground. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\). The balls collide directly. Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) By modelling the balls as particles,

1. show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 5 } u ( 1 + 6 e )\).
   (6) After the collision with ball \(A\), ball \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 5 } { 7 }\) Ball \(B\) rebounds from the wall and there is a second direct collision between \(A\) and \(B\).
2. Find the range of possible values of \(e\).

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction 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 \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). 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 a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

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. 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. 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. 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.

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 \(T = \frac { 112 a } { 15 u }\).

4. A particle \(A\) of mass \(2 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2 u\) 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 \(B\).
2. Show that the kinetic energy lost in the collision is \(7 m u ^ { 2 }\). 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 \(\frac { 14 } { 3 } m u\).
3. Calculate the coefficient of restitution between \(B\) and the wall.
   (4)

6. 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 \(3 m\) 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. 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 \(\frac { 3 } { 4 }\). Given that there is a second collision between \(A\) and \(B\),
2. find the range of values of \(e\) for which the motion described is possible.
   (6) \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a89db1ee-073c-43a3-8480-44970e51c6e2-6_345_1133_440_524}
   \end{figure} A straight \(\log A B\) has weight \(W\) and length \(2 a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 5 } { 12 }\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6 . The log is modelled as a uniform rod and the cable as light.
3. Show that the normal reaction on the \(\log\) at \(A\) is \(\frac { 2 } { 5 } W\).
4. Find the value of \(\beta\). The tension in the cable is \(k W\).
5. Find the value of \(k\).

5. Two small smooth spheres, \(P\) and \(Q\), of equal radius, have masses \(2 m\) and \(3 m\) respectively. The sphere \(P\) is moving with speed \(5 u\) on a smooth horizontal table when it collides directly with \(Q\), which is at rest on the table. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(2 ( 1 + e ) u\). After the collision, \(Q\) hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f , 0 < f \leqslant 1\).
2. Show that, when \(e = 0.4\), there is a second collision between \(P\) and \(Q\). Given that \(e = 0.8\) and there is a second collision between \(P\) and \(Q\),
3. find the range of possible values of \(f\).

8. Two particles \(A\) and \(B\) move on a smooth horizontal table. The mass of \(A\) is \(m\), and the mass of \(B\) is \(4 m\). 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. 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 \(B\),
2. show that \(\frac { 1 } { 4 } < e < \frac { 9 } { 16 }\). Given that \(e = \frac { 1 } { 2 }\),
3. find the total kinetic energy lost in the first collision between \(A\) and \(B\).

7. \begin{figure}[h] \end{figure} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2 m\) 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 \(2 u\) and the speed of \(Q\) is \(3 u\). 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 \(v = 2 w\). 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.
2. Find the coefficient of restitution between \(P\) and \(Q\).

5. A particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal floor. Particle \(A\) collides directly with another particle \(B\) of mass \(2 m\) which is moving along the same straight line with speed \(u\) but in the opposite direction to \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 7 } { 5 } u\)
   2. Find the speed of \(A\) immediately after the collision. 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 \(\frac { 1 } { 2 }\). The first collision between \(A\) and \(B\) occurred at a distance \(x\) from the wall. The particles collide again at a distance \(y\) from the wall.
2. Find \(y\) in terms of \(x\).

8. A particle \(A\) of mass \(m\) is moving with speed \(3 u\) on a smooth horizontal table when it collides directly with a particle \(B\) of mass \(2 m\) which is moving in the opposite direction with speed \(u\). The direction of motion of \(A\) is reversed by the collision. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + 4 e ) u\).
   (6)
2. Show that \(e > \frac { 1 } { 8 }\).
   (3) Subsequently \(B\) hits a wall fixed at right angles to the line of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). After \(B\) rebounds from the wall, there is a further collision between \(A\) and \(B\).
3. Show that \(e < \frac { 1 } { 4 }\).
   (4) END