6.03k Newton's experimental law: direct impact

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 4 kg respectively. The spheres are moving on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres of the spheres, and \(B\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-16_291_844_607_468} Just after the collision, \(B\) moves in a direction perpendicular to the line of centres.

1. Find the speed of \(A\) immediately after the collision.
2. Find the acute angle, correct to the nearest degree, between the velocity of \(A\) and the line of centres immediately after the collision.
3. Find the coefficient of restitution between the spheres.
4. Find the magnitude of the impulse exerted on \(B\) during the collision.

7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).

1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
   Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

3 A disc of mass 0.5 kg is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse received by the disc.
2. Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed \(3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Find the coefficient of restitution between the disc and the wall.
   [0pt] [4 marks]

4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
   2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
3. Explain why it is not necessary to model the spheres as particles in this question.
   [0pt] [2 marks]

5 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\alpha\) to the line of centres of the spheres, and \(B\) has velocity \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\beta\) to the line of centres, as shown in the diagram.
The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 7 }\).
Given that \(\sin \alpha = \frac { 4 } { 5 }\) and \(\sin \beta = \frac { 12 } { 13 }\), find the speeds of \(A\) and \(B\) immediately after the collision.
[0pt] [11 marks]

4 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg , and \(B\) has mass \(m \mathrm {~kg}\). Immediately before the collision, \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, and \(B\) is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(30 ^ { \circ }\) to the line of centres. Immediately after the collision \(A\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(45 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres (see diagram).

1. Find \(u\).
2. Given that \(\theta = 88.1 ^ { \circ }\) correct to 1 decimal place, calculate the approximate values of \(v\) and \(m\).
3. The coefficient of restitution is 0.75 . Show that the exact value of \(\theta\) is a root of the equation \(8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of \(A\) has components \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(B\), and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres. \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\), the component of the velocity of \(A\) along the line of centres immediately after the collision.
2. Given that the speeds of \(A\) and \(B\) are the same immediately after the collision, and that \(3 e ^ { 2 } = 1\), find \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-4_369_953_269_596} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 3 kg and 4 kg respectively. They are moving on a horizontal surface, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. The directions of motion of \(A\) and \(B\) make angles \(\alpha\) and \(\beta\) respectively with the line of centres of the spheres, where \(\sin \alpha = \cos \beta = 0.6\) (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the angle that the velocity of \(A\) makes, immediately after impact, with the line of centres of the spheres.
[0pt] [10]

2 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_421_759_936_694} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and \(B\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres (see diagram). The coefficient of restitution is 0.6 . The direction of motion of \(B\) after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Find the value of \(v\).

4 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-3_497_1157_255_493} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg and \(B\) has mass 0.3 kg . Immediately before the collision \(A\) is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\), and \(B\) is moving with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.7. Find

1. the speed of \(B\) immediately after the collision,
2. the angle turned through by the direction of motion of \(A\) as a result of the collision.

1 A particle \(P\) of mass 0.05 kg is moving on a smooth horizontal surface with speed \(2 \mathrm {~ms} ^ { - 1 }\), when it is struck by a horizontal blow in a direction perpendicular to its direction of motion. The magnitude of the impulse of the blow is \(I\) Ns. The speed of \(P\) after the blow is \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Immediately before the blow \(P\) is moving parallel to a smooth vertical wall. After the blow \(P\) hits the wall and rebounds from the wall with speed \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the coefficient of restitution between \(P\) and the wall.

2 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-2_544_816_781_603} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. They are moving in opposite directions on a horizontal surface and they collide. Immediately before the collision, each sphere has speed \(u \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.5 .

1. Show that the speed of \(B\) is unchanged as a result of the collision.
2. Find the direction of motion of each of the spheres after the collision.

2 Two uniform smooth spheres \(A\) and \(B\), of equal radius and equal mass, are moving towards each other on a horizontal surface. Immediately before they collide, \(A\) has speed \(0.3 \mathrm {~ms} ^ { - 1 }\) along the line of centres and \(B\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_302_1013_1247_502} After the collision, the direction of motion of \(B\) is at right angles to its original direction of motion. Find

1. the speed of \(B\) after the collision,
2. the speed and direction of motion of \(A\) after the collision,
3. the coefficient of restitution between \(A\) and \(B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-3_362_841_264_651} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.

4 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-2_332_995_1375_575} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 3 kg respectively. They are moving on a horizontal surface, and they collide. Immediately before the collision, \(A\) is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the line of centres, where \(\sin \alpha = 0.8\), and \(B\) is moving along the line of centres with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.
[0pt] [10]

7 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-4_588_629_274_758} A particle \(P\) of mass 0.8 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . A particle \(Q\) is suspended from \(O\) by an identical string. With the string \(O P\) taut and inclined at \(\frac { 1 } { 3 } \pi\) radians to the vertical, \(P\) is projected with speed \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the string so as to strike \(Q\) directly (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 7 }\).

1. Calculate the tension in the string immediately after \(P\) is set in motion.
2. Immediately after \(P\) and \(Q\) collide they have equal speeds and are moving in opposite directions. Show that \(Q\) starts to move with speed \(0.15 \mathrm {~ms} ^ { - 1 }\).
3. Prove that before the second collision between \(P\) and \(Q , Q\) is moving with approximate simple harmonic motion.
4. Hence find the time interval between the first and second collisions of \(P\) and \(Q\).

3 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres when it collides with \(B\), which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of \(A\) immediately after the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) after the collision,
2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
3. the value of \(m\),
4. the coefficient of restitution between \(A\) and \(B\). \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
8. the distance travelled by \(P\),
9. the speed of \(P\).

4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find

1. the velocity of \(A\) immediately after the collision,
2. the angle turned through by the direction of motion of \(B\) as a result of the collision.

3 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-2_403_951_1247_559} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.4 kg . Immediately before the collision \(A\) is moving with speed \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision \(A\) is stationary. Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the angle turned through by the direction of motion of \(B\) as a result of the collision. \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

2 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-2_296_798_461_694} A sphere \(A\) of mass \(m\), moving on a horizontal surface, collides with another sphere \(B\) of mass \(2 m\), which is at rest on the surface. The spheres are smooth and uniform, and have equal radius. Immediately before the collision, \(A\) has velocity \(u\) at an angle \(\theta ^ { \circ }\) to the line of centres of the spheres (see diagram). Immediately after the collision, the spheres move in directions that are perpendicular to each other.

1. Find the coefficient of restitution between the spheres.
2. Given that the spheres have equal speeds after the collision, find \(\theta\).

3. A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta , \theta < 45 ^ { \circ }\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.

1. Show that \(e = \tan ^ { 2 } \theta\).

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
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