6.03a Linear momentum: p = mv

5 \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-04_221_1233_367_328} The masses of two spheres \(A\) and \(B\) are \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving towards each other with constant speeds \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 \(v \mathrm {~ms} ^ { - 1 }\) and \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively.

1. Find an expression for \(v\) in terms of \(e\) and \(u\).
2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
   1. the total kinetic energy of the spheres before the collision,
   2. the total kinetic energy of the spheres after the collision.
   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 $$\lambda = \frac { 27 e ^ { 2 } + 25 } { 52 }$$
   4. Comment on the cases when
      (a) \(\lambda = 1\),
      (b) \(\lambda = \frac { 25 } { 52 }\). \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-05_789_981_324_543} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg , by three light rods where the length of rod \(A P\) is 1.5 m and the length of rod \(P Q\) is 0.75 m . Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A , B , P\) and \(Q\) are coplanar. The rod \(A P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical, rod \(P Q\) makes an angle of \(30 ^ { \circ }\) with the downward vertical and rod \(B P\) is horizontal (see diagram).
      1. Find the tension in the \(\operatorname { rod } P Q\).
      2. Find \(\omega\).
      3. Find the speed of \(P\).
      4. Find the tension in the \(\operatorname { rod } A P\).
      5. Hence find the magnitude of the force in rod \(B P\). Decide whether this rod is under tension or compression.

1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}

1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}

1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}

1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}

1 Two particles, \(A\) and \(B\), are travelling in the same direction along a straight line on a smooth horizontal surface. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Particle \(A\) has a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle \(A\) and particle \(B\) collide and coalesce to form a single particle. Find the speed of this single particle after the collision.

2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.

1. Find the velocity of the combined particles after the collision.
2. Find the speed of the combined particles after the collision.

1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.

Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.

9 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

9 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

1. Find the value of \(k\). [4]
2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]

A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). [8]

Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(4u\). In the collision, the particles join together to form a single particle. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision. [6]

A railway truck \(S\) of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck \(T\) of mass 30 tonnes which is at rest. Immediately before the collision the speed of \(S\) is \(4\text{ ms}^{-1}\) As a result of the collision, the two railway trucks join together. Find

1. the common speed of the railway trucks immediately after the collision, [2]
2. the magnitude of the impulse exerted on \(S\) in the collision, stating the units of your answer. [3]

Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m s\(^{-1}\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.

1. Find the value of \(k\). [4]
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]

Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \text{ m s}^{-1}\) and the speed of \(B\) is \(2 \text{ m s}^{-1}\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find

1. the speed of \(A\) immediately after the collision, [5]
2. the magnitude of the impulse exerted on \(B\) in the collision. [3]

Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2m\) and the mass of \(B\) is \(m\). The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of \(A\) is \(2u\) and the speed of \(B\) is \(3u\). The magnitude of the impulse received by each particle in the collision is \(\frac{7mu}{2}\). Find

1. the speed of \(A\) immediately after the collision, [3]
2. the speed of \(B\) immediately after the collision. [3]

A particle \(P\) of mass 0.25 kg moves under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (2 - 4t)\mathbf{i} + (t^2 + 2t)\mathbf{j}$$ When \(t = 0\), \(P\) is at the point with position vector \((2\mathbf{i} - 4\mathbf{j})\) m with respect to a fixed origin \(O\). When \(t = 3\), \(P\) is at the point \(A\). Find

1. the momentum of \(P\) when \(t = 3\), [2]
2. the magnitude of \(\mathbf{F}\) when \(t = 3\), [6]
3. the position vector of \(A\). [5]

\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses \(0.8\) kg, \(0.6\) kg and \(0.7\) kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4\) m s\(^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2\) m s\(^{-1}\) and \(0.5\) m s\(^{-1}\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2\) m s\(^{-1}\). Find the speed of \(L\) after the collision. [4]
2. \(M\) then collides with \(N\).
   1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
   2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]

\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses 0.8 kg, 0.6 kg and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \text{ m s}^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \text{ m s}^{-1}\) and \(0.5 \text{ m s}^{-1}\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \text{ m s}^{-1}\). Find the speed of \(L\) after the collision. [4]
2. \(M\) then collides with \(N\).
   1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
   2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]

Two particles \(P\) and \(Q\), of mass \(m\) and \(km\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3u\) and \(2u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.

1. Find the value of \(k\). [4 marks]
2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision. [3 marks]

Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]

Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).

1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]

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

Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 ms\(^{-1}\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.

1. Show that the speed of \(B\) after the impact cannot be more than 3 ms\(^{-1}\). [5 marks]

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(e < \frac{3}{10}\). [5 marks]
2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]

\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).

1. Find the coefficient of restitution between \(A\) and \(B\). [4]
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]

Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.

1. Show that there will be another collision. [3]