Direct collision of particles

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(A\) has mass 2 kg and a particle \(B\) has mass 3 kg . The particles are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocity of \(A\) is \(5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(B\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse received by \(A\) in the collision. Given that, in the collision, the impulse of \(A\) on \(B\) is equal and opposite to the impulse of \(B\) on \(A\),
3. find the velocity of \(B\) immediately after the collision.

1 Two uniform small smooth spheres, \(A\) and \(B\), of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere \(A\) receives an impulse of magnitude 8 N s in the direction \(A B\). The coefficient of restitution between the spheres is \(e\). Find, in terms of \(e\), the speeds of \(A\) and \(B\) after \(A\) collides with \(B\). Given that the spheres move in opposite directions after the collision, show that \(e > \frac { 2 } { 3 }\).

3 Two identical uniform small spheres \(A\) and \(B\), each of mass \(m\), are moving towards each other in a straight line on a smooth horizontal surface. Their speeds are \(u\) and \(k u\) respectively, and they collide directly. The coefficient of restitution between the spheres is \(e\). Sphere \(B\) is brought to rest by the collision.

1. Show that \(e = \frac { k - 1 } { k + 1 }\).
2. Given that \(60 \%\) of the total initial kinetic energy is lost in the collision, find the values of \(k\) and \(e\).

5 Two particles, \(A\) of mass \(m _ { A } \mathrm {~kg}\) and \(B\) of mass 5 kg , are moving directly towards each other on a smooth horizontal floor. Before they collide they have speeds \(\mathrm { u } _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after they collide the direction of motion of each particle has been reversed and \(A\) and \(B\) have speeds \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.75 . Before:
\includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_711_552_283} After:
\includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_707_552_1078}

1. Determine the value of \(m _ { A }\) and the value of \(u _ { A }\).
   [0pt] [5]
2. Show that approximately \(41 \%\) of the kinetic energy of the system is lost in this collision. After the collision between \(A\) and \(B\), \(B\) goes on to collide directly with a third particle \(C\) of mass 3 kg which is travelling towards \(B\) with a speed of \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(B\) and \(C\) is denoted by \(e\).
3. Given that, after \(B\) and \(C\) collide, there are no further collisions between \(A , B\) and \(C\) determine the range of possible values of \(e\).

5
\includegraphics[max width=\textwidth, alt={}, center]{4799bae3-8d1e-4ca7-8621-6fa195469175-4_225_1239_367_324} 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
   (a) the total kinetic energy of the spheres before the collision,
   (b) 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]{4799bae3-8d1e-4ca7-8621-6fa195469175-5_808_990_319_539} 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).
5. Find the tension in the \(\operatorname { rod } P Q\).
6. Find \(\omega\).
7. Find the speed of \(P\).
8. Find the tension in the rod \(A P\).
9. Hence find the magnitude of the force in rod \(B P\). Decide whether this rod is under tension or compression. OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{}

7
\includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.

6 The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(10 \mathrm {~ms} ^ { - 1 }\) before they collide. The kinetic energy lost due to the collision is 121.5 J .

1. Find the speed and direction of motion of each particle after the collision.
2. Find the coefficient of restitution between \(A\) and \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle \(P\) is projected with speed \(32 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac { 24 } { 25 }\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
3. Calculate the height of \(C\) above the ground and the distance \(A B\). Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
4. Given that the mass of \(P\) is 3 kg , find the magnitude and direction of the impulse exerted on \(P\) by the ground. The coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\).
5. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25 ^ { \circ }\) below the horizontal.

1 Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85 . \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with A . In this collision the coefficient of restitution is 0.4 , the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.
2. Show that the velocity of A immediately after the collision is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slope. Find the velocity of B immediately after the collision.
3. Calculate the impulse on B in the collision. The blocks do not collide again.
4. For what length of time after the collision does A slide before it comes to rest?

2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).

1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).

4 A smooth sphere \(A\), of mass \(m\), is moving with speed \(4 u\) in a straight line on a smooth horizontal table. A smooth sphere \(B\), of mass \(3 m\), has the same radius as \(A\) and is moving on the table with speed \(2 u\) in the same direction as \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625} The sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speeds of \(A\) and \(B\) immediately after the collision.
2. Show that the speed of \(B\) after the collision cannot be greater than \(3 u\).
3. Given that \(e = \frac { 2 } { 3 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) in the collision.

5. \begin{figure}[h] \end{figure} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg . After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m , and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m . The angle \(A X B\) is \(90 ^ { \circ }\). The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van.
   (5) The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(A X\) and the car to a constant horizontal force of 300 N along \(B X\).
2. Find the speed of the car immediately before the collision.
   (9)

2. Two small smooth spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(2 m \mathrm {~kg}\) and the mass of \(B\) is \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of \(A\) is \(( 2 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the speed of \(B\) immediately after the collision.
(5)

5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .

1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.

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
   (a) the total kinetic energy of the spheres before the collision,
   (b) 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).
5. Find the tension in the \(\operatorname { rod } P Q\).
6. Find \(\omega\).
7. Find the speed of \(P\).
8. Find the tension in the \(\operatorname { rod } A P\).
9. Hence find the magnitude of the force in rod \(B P\). Decide whether this rod is under tension or compression.

4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4

1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
2. Find the speed of \(A\) in terms of \(u\) and \(e\).
   4
3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
   4
4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
   Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$

14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\)
Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks]
\(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5

1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\)
   [0pt] [5 marks]
   5
2. The solid is suspended in equilibrium from a point on the edge of the circular face.
   Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
   6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
   The other end of the string is attached to a fixed point \(O\)
   The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
3. Find the speed of the sphere when it is directly below \(O\)
   6
4. State one assumption that you made about the string.
   6
5. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
6. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
   Explain why this result is reasonable, showing any calculations that you make.
   7 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle \(A\) has mass 2 kg and moves with velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line. Particle \(B\) has mass 3 kg and moves with velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) to the line. The particles and their velocities are shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-08_451_739_858_653} The particles collide when they reach the line and then move together as a single combined particle. 7
7. Show that the magnitude of the impulse on particle \(A\) during the collision is 7.55 Ns correct to three significant figures.
   7
8. State the magnitude of the impulse on \(B\) during the collision, giving a reason for your answer. 7
9. Find the size of the angle between the straight line and the impulse acting on \(B\), giving your answer to the nearest degree. 7
10. During the collision, one particle crosses the straight line.
    State which particle crosses the line, giving a reason for your answer.
    [0pt] [1 mark] 8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A block has mass 10 kg and is at rest 1 metre from a fixed point \(O\) on a horizontal surface. One end of an elastic string is attached to the block and the other end of the elastic string is attached to the point \(O\) The elastic string has modulus of elasticity 40 newtons and natural length 2 metres.
    The coefficient of friction between the block and the surface is 0.6 A force is applied to the block so that it starts to move towards a vertical wall.
    The block moves on a line that is perpendicular to the wall.
    The force has magnitude 100 newtons and acts at an angle of \(30 ^ { \circ }\) to the horizontal.
    The situation is shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-10_239_1339_1176_354} 8
11. Show that the distance that the block has moved, when the forces acting on it are in equilibrium, is 3.9 metres correct to two significant figures.
    [0pt] [4 marks]
    8
12. State one limitation of the model that you have used. 8
13. Find the maximum speed of the block.
    8
14. The vertical wall is 8.7 metres from \(O\) Determine whether the block reaches the wall, showing any calculations that you make.
    \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-13_2492_1721_217_150}

10 A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(Q\) of mass 8 kg is released from rest 5 m vertically above P . During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac { 11 } { 14 }\). Determine the time between this collision and P subsequently hitting the ground.