Direct collision with speed relationships

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.
   

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.

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 Two spheres \(A\) and \(B\) are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are 2 kg and 3 kg respectively.
Both \(A\) and \(B\) are initially at rest.
Sphere \(A\) is set in motion directly towards sphere \(B\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and subsequently collides with sphere \(B\) The coefficient of restitution between the spheres is \(e\)
8

1. 1. Show that the speed of \(B\) immediately after the collision is $$\frac { 8 ( 1 + e ) } { 5 }$$ 8
2. (ii) Find an expression, in terms of \(e\), for the velocity of \(A\) immediately after the collision.
   8
3. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\)
   [0pt] [2 marks]
   8
4. 1. The impulse of sphere \(A\) on sphere \(B\) is \(I\)
      The impulse of sphere \(B\) on sphere \(A\) is \(J\)
      Given that the collision is perfectly inelastic, find the value of \(I + J\)
      8
5. (ii) State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic.
   \includegraphics[max width=\textwidth, alt={}, center]{a12155cc-cd07-40e0-af69-6b2590e4ea7c-12_2488_1732_219_139}
   \includegraphics[max width=\textwidth, alt={}]{a12155cc-cd07-40e0-af69-6b2590e4ea7c-16_2496_1721_214_148}

5 Two small smooth discs, \(C\) and \(D\), have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc \(C\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and disc \(D\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they collide. The coefficient of restitution between \(C\) and \(D\) is 0.6 The diagram shows the discs, viewed from above, before the collision.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

1. Show that the speed of \(D\) immediately after the collision is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 significant figures.
   5
2. Find the speed of \(C\) immediately after the collision.
   [0pt] [2 marks]
   5
3. In fact the horizontal surface on which the discs are sliding is not smooth.
   Explain how the introduction of friction will affect your answer to part (b).
   [0pt] [2 marks]
   Turn over for the next question

3 A particle \(A\) of mass 0.5 kg is moving with a speed of \(3.15 \mathrm {~ms} ^ { - 1 }\) on a smooth horizontal surface when it collides directly with a particle \(B\) of mass 0.8 kg which is at rest on the surface. The velocities of \(A\) and \(B\) immediately after the collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { v } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. You are given that \(\mathrm { v } _ { \mathrm { B } } = 2 \mathrm { v } _ { \mathrm { A } }\).

1. Find the values of \(\mathrm { V } _ { \mathrm { A } }\) and \(\mathrm { V } _ { \mathrm { B } }\).
2. Find the coefficient of restitution between \(A\) and \(B\).
3. Explain why the coefficient of restitution is a dimensionless quantity.
4. Calculate the total loss of kinetic energy as a result of the collision.
5. State, giving a reason, whether or not the collision is perfectly elastic.
6. Calculate the impulse that \(B\) exerts on \(A\) in the collision.

5. Two railway trucks \(A\) and \(B\), whose masses are \(6 m\) and \(5 m\) respectively, are moving in the same direction along a straight track with speeds \(5 u\) and \(3 u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(k v\) 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. \(v = \frac { 45 u } { 5 k + 6 }\),
   2. \(v = \frac { 2 e u } { k - 1 }\).
      (8 marks)
2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). \section*{MECHANICS 2 (A) TEST PAPER 7 Page 2}

6
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_243_1179_1580_443} The masses of two particles \(A\) and \(B\) are 0.2 kg and \(m \mathrm {~kg}\) respectively. The particles are moving with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(u \mathrm {~ms} ^ { - 1 }\) in the same horizontal line and in the same direction (see diagram). The two particles collide and the coefficient of restitution between the particles is \(e\). After the collision, \(A\) and \(B\) continue in the same direction with speeds \(4 \left( 1 - e + e ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.

1. Find \(u\) and \(m\) in terms of \(e\).
2. Find the value of \(e\) for which the speed of \(A\) after the collision is least and find, in this case, the total loss in kinetic energy due to the collision.
3. Find the possible values of \(e\) for which the magnitude of the impulse that \(B\) exerts on \(A\) is 0.192 Ns .
   \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-5_744_887_264_589} The diagram shows a surface consisting of a horizontal part \(O A\) and a plane \(A B\) inclined at an angle of \(70 ^ { \circ }\) to the horizontal. A particle is projected from the point \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal \(O A\). The particle hits the plane \(A B\) at the point \(P\), with speed \(14 \mathrm {~ms} ^ { - 1 }\) and at right angles to the plane, 1.4 s after projection.
4. Show that the value of \(u\) is 15.9 , correct to 3 significant figures, and find the value of \(\theta\).
5. Find the height of \(P\) above the level of \(A\). The particle rebounds with speed \(v \mathrm {~ms} ^ { - 1 }\). The particle next lands at \(A\).
6. Find the value of \(v\).
7. Find the coefficient of restitution between the particle and the plane at \(P\).

1

1. Two model railway trucks are moving freely on a straight horizontal track when they are in a direct collision. The trucks are P of mass 0.5 kg and Q of mass 0.75 kg . They are initially travelling in the same direction. Just before they collide P has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and Q has a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_263_640_484_715} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Suppose that the speed of P is halved in the collision and that its direction of motion is not changed. Find the speed of Q immediately after the collision and find the coefficient of restitution.
   2. Show that it is not possible for both the speed of P to be halved in the collision and its direction of motion to be reversed. Both of the model trucks have flat horizontal tops. They are each travelling at the speeds they had immediately after the collision. Part of the mass of Q is a small object of mass 0.1 kg at rest at the edge of the top of Q nearest P . The object falls off, initially with negligible velocity relative to Q .
   3. Determine the speed of Q immediately after the object falls off it, making your reasoning clear. Part of the mass of P is an object of mass 0.05 kg that is fired horizontally from the top of P , parallel to and in the opposite direction to the motion of P . Immediately after the object is fired, it has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to P .
   4. Determine the speed of P immediately after the object has been fired from it.
2. The velocities of a small object immediately before and after an elastic collision with a horizontal plane are shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_172_741_1987_644} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Show that the plane cannot be smooth.

6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),

1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the time taken for \(T\) to reach \(X\).
3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
   (4 marks)

7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),

1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. find \(e\),
3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.

2
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-02_232_609_840_239} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and \(B\) is moving away from \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the possible speeds with which B moves after the collision.