6.03f Impulse-momentum: relation

5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.

1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).

2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.

1. Calculate the magnitude of the impulse which the ground exerts on the sphere.
2. Calculate the coefficient of restitution between the sphere and the ground.

5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

6 A small ball \(B\) is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) from a point \(O\) on a horizontal plane, and moves freely under gravity.

1. Calculate the height of \(B\) above the plane when moving horizontally. \(B\) has mass 0.4 kg . At the instant when \(B\) is moving horizontally it receives an impulse of magnitude \(I \mathrm { Ns }\) in its direction of motion which immediately increases the speed of \(B\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate \(I\). For the instant when \(B\) returns to the plane, calculate
3. the speed and direction of motion of \(B\),
4. the time of flight, and the distance of \(B\) from \(O\).

6 A small ball of mass 0.5 kg is held at a height of 3.136 m above a horizontal floor. The ball is released from rest and rebounds from the floor. The coefficient of restitution between the ball and floor is \(e\).

1. Find in terms of \(e\) the speed of the ball immediately after the impact with the floor and the impulse that the floor exerts on the ball. The ball continues to bounce until it eventually comes to rest.
2. Show that the time between the first bounce and the second bounce is \(1.6 e\).
3. Write down, in terms of \(e\), the time between
   1. the second bounce and the third bounce,
   2. the third bounce and the fourth bounce.
   3. Given that the time from the ball being released until it comes to rest is 5 s , find the value of \(e\).

2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds.

2 The graph shows how a force, \(F\), varies with time during a period of 0.8 seconds. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of \(F\) during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, \(F\) newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where \(k\) and \(b\) are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

1. State the value of \(b\).
   7
2. Find the magnitude of the impulse on the disc, giving your answer in terms of \(k\).
   7
3. The disc is travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the wall.
   The disc rebounds with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find \(k\).
   [0pt] [3 marks]

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\). \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

2 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg . \(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the speed of \(A\) after the string becomes taut.
2. Find the impulse exerted on \(A\) at the instant that the string becomes taut.
3. Find the loss in kinetic energy as a result of the string becoming taut.

2 A hockey puck of mass 0.2 kg is sliding down a rough slope which is inclined at \(10 ^ { \circ }\) to the horizontal. At the instant that its velocity is \(14 \mathrm {~ms} ^ { - 1 }\) directly down the slope it is hit by a hockey stick. Immediately after it is hit its velocity is \(24 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. Find the magnitude of the impulse exerted by the hockey stick on the puck. After it has been hit, the puck first comes to instantaneous rest when it has travelled 15 m up the slope. While the puck is moving up the slope, the resistance to its motion has constant magnitude \(R \mathrm {~N}\).
2. Use an energy method to determine the value of \(R\).

1 Two particles \(A\), of mass \(m \mathrm {~kg}\), and \(B\), of mass \(3 m \mathrm {~kg}\), are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. \(A\) is projected along the surface, directly away from \(B\), with a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(B\) immediately after the string becomes taut.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) as a result of the string becoming taut.

7 A particle \(P\) of mass 3.5 kg is attached to one end of a rod of length 5.4 m . The other end of the rod is hinged at a fixed point \(O\) and \(P\) hangs in equilibrium directly below \(O\). A horizontal impulse of magnitude 44.1 Ns is applied to \(P\).
In an initial model of the subsequent motion of \(P\) the rod is modelled as being light and inextensible and all resistance to the motion of \(P\) is ignored. You are given that \(P\) moves in a circular path in a vertical plane containing \(O\). The angle that the rod makes with the downward vertical through \(O\) is \(\theta\) radians.

1. Determine the largest value of \(\theta\) in the subsequent motion of \(P\). In a revised model the rod is still modelled as being light and inextensible but the resistance to the motion of \(P\) is not ignored. Instead, it is modelled as causing a loss of energy of 20 J for every metre that \(P\) travels.
2. Show that according to the revised model, the maximum value of \(\theta\) in the subsequent motion of \(P\) satisfies the following equation. $$343 ( 1 + 2 \cos \theta ) = 400 \theta$$ You are given that \(\theta = 1.306\) is the solution to the above equation, correct to \(\mathbf { 4 }\) significant figures.
3. Determine the difference in the predicted maximum vertical heights attained by \(P\) using the two models. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. Suggest one further improvement that could be made to the model of the motion of \(P\).

6 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

3 A particle \(P\) of mass 6 kg moves in a straight line under the action of a single force of magnitude \(F N\) which acts in the direction of motion of \(P\).
At time \(t\) seconds, where \(t \geqslant 0 , F\) is given by \(\mathrm { F } = \frac { 1 } { 5 - 4 \mathrm { e } ^ { - \mathrm { t } ^ { 2 } } }\).
When \(t = 0\), the speed of \(P\) is \(1.9 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse of the force over the period \(0 \leqslant t \leqslant 2\).
2. Find the speed of \(P\) at the instant when \(t = 2\).
3. Find the work done by the force on \(P\) over the period \(0 \leqslant t \leqslant 2\).

6 A particle \(P\) of mass 2.5 kg is free to move along the \(x\)-axis. When its displacement from the origin is \(x \mathrm {~m}\) its velocity is \(v \mathrm {~ms} ^ { - 1 }\). At time \(t = 0\) seconds, \(P\) is at the point where \(x = 1\) and is travelling in the negative \(x\)-direction with speed \(5 \mathrm {~ms} ^ { - 1 }\). At this time an impulse of \(I\) Ns is applied to \(P\) in the positive \(x\)-direction so that \(P\) moves in the positive \(x\)-direction with speed \(18 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Subsequently, whenever \(P\) is in motion, two forces act on it. The first force acts in the positive \(x\)-direction and has magnitude \(\frac { 5 v ^ { 2 } } { x } N\). The second force acts in the negative \(x\)-direction and has magnitude 60 vN .
2. Show that the motion of \(P\) can be modelled by the differential equation \(\frac { \mathrm { dV } } { \mathrm { dx } } = \frac { \mathrm { aV } } { \mathrm { x } } + \mathrm { b }\) where \(a\) and \(b\) are constants whose values should be determined.
3. By solving the differential equation derived in part (b) find an expression for \(v\) in terms of \(x\). You are given that \(\mathrm { x } = \frac { 4 } { 3 \mathrm { e } ^ { - 24 \mathrm { t } } + 1 }\) when \(t \geqslant 0\).
4. Describe in detail the motion of \(P\) when \(t \geqslant 0\).

1 A particle \(P\) of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall. At the instant before the collision, the velocity of \(P\) is \(- 5 \mathbf { i } + 12 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
At the instant after the collision, the velocity of \(P\) is \(\mathbf { i } + 4 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find the magnitude of the momentum of \(P\) before the collision.
2. Find, in vector form, the impulse that the wall exerts on \(P\).
3. State, in vector form, the impulse that \(P\) exerts on the wall.
4. Find in either order.

2 A particle \(P\) of mass 2 kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is \(( 5 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision its velocity is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. The impulse imparted on \(P\) by the wall is denoted by INs. Find the following.

6 A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string of length 0.8 m . The other end of the string is attached to a fixed point \(O . P\) is at rest vertically below \(O\) when it experiences a horizontal impulse of magnitude 20 Ns . In the subsequent motion the angle the string makes with the downwards vertical through \(O\) is denoted by \(\theta\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-4_387_502_1434_255}

1. Find the magnitude of the acceleration of \(P\) at the first instant when \(\theta = \frac { 1 } { 3 } \pi\) radians.
2. Determine the value of \(\theta\) at which the string first becomes slack.

3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F \mathrm {~N}\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).

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}

7. Two particles \(A\) and \(B\), of mass \(3 M \mathrm {~kg}\) and \(2 M \mathrm {~kg}\) respectively, are moving towards each other on a rough horizontal track. Just before they collide, \(A\) has speed \(3 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(5 \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, the direction of motion of both particles has been reversed and they are both travelling at the same speed, \(v\).

1. Show that \(v = 1 \mathrm {~ms} ^ { - 1 }\). The magnitude of the impulse exerted on \(A\) during the collision is 24 Ns.
2. Find the value of \(M\). Given that the coefficient of friction between \(A\) and the track is 0.1 ,
3. find the time taken from the moment of impact until \(A\) comes to rest. END

5. A cricket ball of mass 0.3 kg is approaching a batsman at \({ } ^ { - } 30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The batsman hits the ball with a 1.5 kg bat moving with velocity \(15 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Contact between bat and ball lasts for 0.2 seconds. Immediately after this, bat and ball move with velocities \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and \(v \mathbf { i } \mathrm {~ms} ^ { - 1 }\) respectively.

1. Suggest a suitable model for the cricket ball.
2. Calculate the value of \(v\).
3. Find the magnitude of the force with which the batsman hits the ball.