Collision with two possible outcomes

1 Particles \(P\) of mass 0.4 kg and \(Q\) of mass 0.5 kg are free to move on a smooth horizontal plane. \(P\) and \(Q\) are moving directly towards each other with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. After \(P\) and \(Q\) collide, the speed of \(Q\) is twice the speed of \(P\). Find the two possible values of the speed of \(P\) after the collision.

1 Two particles \(P\) and \(Q\), of masses 0.1 kg and 0.4 kg respectively, are free to move on a smooth horizontal plane. Particle \(P\) is projected with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After \(P\) and \(Q\) collide, the speeds of \(P\) and \(Q\) are equal. Find the two possible values of the speed of \(P\) after the collision.

4 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and \(m \mathrm {~kg}\) respectively, lie on a smooth horizontal plane. Initially, sphere \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collision \(A\) moves with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the two possible values of the loss of kinetic energy due to the collision.

4 Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8 \mathrm {~ms} ^ { - 1 }\). After \(A\) and \(B\) collide, \(A\) moves with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the greater of the two possible total losses of kinetic energy due to the collision.
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-06_2722_43_107_2004}
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-07_197_1142_254_460} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string.The string is at an angle of \(30 ^ { \circ }\) above the plane and has tension \(T \mathrm {~N}\)(see diagram).The coefficient of friction between the particle and the plane is 0.5 .

1. Given that the particle is on the point of moving,find the value of \(T\) .
2. Given instead that the particle is accelerating at \(0.2 \mathrm {~ms} ^ { - 2 }\) ,find the value of \(T\) .

3. Two particles \(A\) and \(B\) have mass \(2 m\) and \(k m\) respectively. The particles are moving in opposite directions along the same straight smooth horizontal line so that the particles collide directly. Immediately before the collision \(A\) has speed \(2 u\) and \(B\) has speed \(u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { u } { 2 }\).

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(B\) on \(A\) in the collision.
2. Show that \(k < 5\)

1. Two particles, \(P\) and \(Q\), of mass \(m _ { 1 }\) and \(m _ { 2 }\) respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed \(u\).

The direction of motion of each particle is reversed by the collision.
Immediately after the collision, the speed of \(Q\) is \(\frac { 1 } { 3 } u\).

1. Find, in terms of \(m _ { 2 }\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision.
2. Find, in terms of \(m _ { 1 } , m _ { 2 }\) and \(u\), the speed of \(P\) immediately after the collision.
3. Hence show that \(m _ { 2 } > \frac { 3 } { 4 } m _ { 1 }\)

2. Two particles, \(A\) and \(B\), are moving in a straight line in opposite directions towards each other on a smooth horizontal surface when they collide directly. Particle \(A\) has mass \(3 m \mathrm {~kg}\) and particle \(B\) has mass \(m \mathrm {~kg}\).
Immediately before the collision, both particles have a speed of \(1.5 \mathrm {~ms} ^ { - 1 }\)
Immediately after the collision, the direction of motion of \(A\) is unchanged and the difference between the speed of \(A\) and speed of \(B\) is \(1 \mathrm {~ms} ^ { - 1 }\)

1. Find (i) the speed of \(A\) immediately after the collision,
   (ii) the speed of \(B\) immediately after the collision.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) in the collision.

2. Particle \(A\) of mass \(2 m\) and particle \(B\) of mass \(k m\), where \(k\) is a positive constant, are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of \(A\) is \(u\) and the speed of \(B\) is \(3 u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { 1 } { 2 } u\).

1. Show that \(k < 1\)
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by \(A\) in the collision.

3. Two particles, \(P\) and \(Q\), have masses 0.5 kg and \(m \mathrm {~kg}\) 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 \(4 \mathrm {~ms} ^ { - 1 }\) and the speed of \(Q\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is 4.2 N s . As a result of the collision the direction of motion of \(P\) is reversed.

1. Find the speed of \(P\) immediately after the collision. The speed of \(Q\) immediately after the collision is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the two possible values of \(m\).

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

Immediately before the collision the speed of \(P\) is \(k u\), where \(k\) is a constant, and the speed of \(Q\) is \(2 u\). Immediately after the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\).
The direction of motion of \(Q\) is reversed by the collision.

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
2. Find the two possible values of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-03_2647_1837_118_114}

3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(k m\), where \(2 < k < 3\), 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 \(2 u\) and the speed of \(B\) is \(4 u\). As a result of the collision the speed of \(A\) is halved and its direction of motion is reversed.

1. Find, in terms of \(k\) and \(u\), the speed of \(B\) immediately after the collision.
2. State whether the direction of motion of \(B\) changes as a result of the collision, explaining your answer. Given that \(k = \frac { 7 } { 3 }\),
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse that \(A\) exerts on \(B\) in the collision.

1
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_227_878_269_635} A particle \(P\) of mass 0.5 kg is travelling with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane towards a stationary particle \(Q\) of mass \(m \mathrm {~kg}\) (see diagram). The particles collide, and immediately after the collision \(P\) has speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that both particles are moving in the same direction after the collision, calculate \(m\).
2. Given instead that the particles are moving in opposite directions after the collision, calculate \(m\).

4
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_149_606_1626_772} Two particles of masses 0.18 kg and m kg move on a smooth horizontal plane. They are moving towards each other in the same straight line when they collide. Immediately before the impact the speeds of the particles are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. Given that the particles are brought to rest by the impact, find m .
2. Given instead that the particles move with equal speeds of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after the impact, find
   (a) the value of m , assuming that the particles move in opposite directions after the impact,
   (b) the two possible values of m , assuming that the particles coalesce.

2 Two particles \(P\) and \(Q\) are moving in opposite directions in the same straight line on a smooth horizontal surface when they collide. \(P\) has mass 0.4 kg and speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 } . Q\) has mass 0.6 kg and speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the speed of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that \(P\) and \(Q\) are moving in the same direction after the collision, find the speed of \(Q\).
2. Given instead that \(P\) and \(Q\) are moving in opposite directions after the collision, find the distance between them 3 s after the collision.

4 An object is projected vertically upwards with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate

1. the speed of the object when it is 2.1 m above the point of projection,
2. the greatest height above the point of projection reached by the object,
3. the time after projection when the object is travelling downwards with speed \(5.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-3_227_897_635_664} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A particle \(P\) of mass 0.5 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface towards a stationary particle \(Q\) of mass \(m \mathrm {~kg}\) (see Fig. 1). After the particles collide, \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in its original direction of motion, and \(Q\) has speed \(1 \mathrm {~ms} ^ { - 1 }\) more than \(P\). Show that \(v ( m + 0.5 ) = - m + 3\).
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-3_229_901_1265_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(Q\) and \(P\) are now projected towards each other with speeds \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see Fig. 2). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with its direction of motion unchanged and \(P\) has speed \(1 \mathrm {~ms} ^ { - 1 }\) more than \(Q\). Find another relationship between \(m\) and \(v\) in the form \(v ( m + 0.5 ) = a m + b\), where \(a\) and \(b\) are constants.
5. By solving these two simultaneous equations show that \(m = 0.9\), and hence find \(v\).

4
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles \(P\) and \(Q\) are moving towards each other with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) along the same straight line on a smooth horizontal surface (see diagram). \(P\) has mass 0.2 kg and \(Q\) has mass 0.3 kg . The two particles collide.

1. Show that \(Q\) must change its direction of motion in the collision.
2. Given that \(P\) and \(Q\) move with equal speed after the collision, calculate both possible values for their speed after they collide.

1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.5 kg respectively are at rest on a smooth horizontal plane. Particle \(P\) is projected with a speed \(6 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(1 \mathrm {~ms} ^ { - 1 }\). Find the two possible speeds of \(Q\) after the collision.
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-02_2716_35_143_2012}

5 Two particles, \(A\) and \(B\), are moving towards each other along the same straight horizontal line when they collide. Particle \(A\) has mass 5 kg and particle \(B\) has mass 4 kg . Just before the collision, the speed of \(A\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision, the speed of \(A\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and both particles move on the same straight horizontal line. Find the two possible speeds of \(B\) after the collision.
(6 marks)

8 Two particles, \(A\) and \(B\), are travelling towards each other along a straight horizontal line.
Particle \(A\) has velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass \(m \mathrm {~kg}\).
Particle \(B\) has velocity \(- 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass 3 kg .
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-5_220_1157_516_440} The particles collide.

1. If the particles move in opposite directions after the collision, each with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(m\).
2. If the particles coalesce during the collision, forming a single particle which moves with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the two possible values of \(m\).

4. Two particles \(A\) and \(B\), of masses 50 grams and \(y\) grams, are moving in the same straight line, in opposite directions, with speeds \(7 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively, and collide.
\includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_218_508_2143_1382}
In each of the following separate cases, find the value of \(y\) and the magnitude of the impulse exerted by each particle on the other:

1. after impact the particles move together with speed \(2.25 \mathrm {~ms} ^ { - 1 }\);
2. after impact the particles move in opposite directions with speed \(5 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 1 (A) TEST PAPER 6 Page 2}

2. The masses of two particles \(A\) and \(B\) are 0.5 kg and \(m \mathrm {~kg}\) respectively. The particles are moving on a smooth horizontal table in opposite directions and collide directly. Immediately before the collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision, the magnitude of the impulse exerted by \(B\) on \(A\) is 3.6 Ns. As a result of the collision the direction of motion of \(A\) is reversed.

1. Find the speed of \(A\) immediately after the collision. The speed of \(B\) immediately after the collision is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the two possible values of \(m\). \section*{3.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d23dfb3f-2969-4bca-a369-1d51e6ba0052-3_250_805_397_644}
   \end{figure} A uniform rod \(A B\) has length 100 cm . Two light pans are suspended, one from each end of the rod, by two strings which are assumed to be light and inextensible. The system forms a balance with the rod resting horizontally on a smooth pivot, as shown in Fig. 1. A particle of weight 16 N is placed in the pan at \(A\) and a particle of weight 5 N is placed in the pan at \(B\). The rod rests horizontally in equilibrium when the pivot is at the point \(C\) on the rod, where \(A C = 30 \mathrm {~cm}\).
3. Find the weight of the rod.
   (3) The particle in the pan at \(A\) is replaced by a particle of weight 3.5 N . The particle of weight 5 N remains in the pan at \(B\). The rod now rests horizontally in equilibrium when the pivot is moved to the point \(D\).
4. Find the distance \(A D\).
5. Explain briefly where the assumption that the strings are light has been used in your answer to part (a).