Coalescence or perfectly inelastic collision

2 Two smooth spheres \(A\) and \(B\), of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with \(B\), which is stationary. The collision is perfectly elastic. Calculate the speed of \(A\) after the impact. [4]

1 A sledge and a child sitting on it have a combined mass of 29.5 kg . The sledge slides on horizontal ice with negligible resistance to its movement.

1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution in the collision is 0.8 . After the impact the speeds of the sledge and the ball are \(V _ { 1 } \mathrm {~ms} ^ { - 1 }\) and \(V _ { 2 } \mathrm {~ms} ^ { - 1 }\) respectively. Calculate \(V _ { 1 }\) and \(V _ { 2 }\) and state the direction in which the ball is travelling after the impact. [7]
2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The snowball sticks to the sledge.
   (A) Calculate the velocity with which the combined sledge and snowball start to move.
   (B) The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer.
3. In another situation, the sledge is travelling over the ice at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with 10.5 kg of snow on it (giving a total mass of 40 kg ). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the snowball and sledge are separating at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-3_527_720_335_667} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\), \(\mathrm { BD } , \mathrm { BE } , \mathrm { CE }\) and DE . [The triangles \(\mathrm { ABD } , \mathrm { BDE }\) and BCE are all equilateral.] The rods \(\mathrm { AB } , \mathrm { BC }\) and DE are horizontal.
   The rods are freely pin-jointed to each other at A, B, C, D and E.
   The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD . The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(L \mathrm {~N}\) at C ; the normal reaction force \(R \mathrm {~N}\) of the plane on the framework at D ; the horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), respectively, acting at A .
4. Write down equations for the horizontal and vertical equilibrium of the framework.
5. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt { 3 } L\) and \(Y = 0\).
6. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods.
7. Show that the internal force in the rod AD is zero.
8. Find the forces internal to \(\mathrm { AB } , \mathrm { CE }\) and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.]
9. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust.

4

1. A large nail of mass 0.02 kg has been driven a short distance horizontally into a fixed block of wood, as shown in Fig. 4.1, and is to be driven horizontally further into the block. The wood produces a constant resistance of 2.43 N to the motion of the nail. The situation is modelled by assuming that linear momentum is conserved when the nail is struck, that all the impacts with the nail are direct and that the head of the nail never reaches the wood. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_279_711_482_676} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The nail is first struck by an object of mass 0.1 kg that is moving parallel to the nail with linear momentum of magnitude 0.108 Ns . The object becomes firmly attached to the nail.
   1. Calculate the speed of the nail and object immediately after the impact.
   2. Calculate the time for which the nail and object move, and the distance they travel in that time. On a second attempt to drive in the nail, it is struck by the same object of mass 0.1 kg moving parallel to the nail with the same linear momentum of magnitude 0.108 Ns . This time the object does not become attached to the nail and after the contact is still moving parallel to the nail. The coefficient of restitution in the impact is \(\frac { 1 } { 3 }\).
   3. Calculate the speed of the nail immediately after this impact.
2. A small ball slides on a smooth horizontal plane and bounces off a smooth straight vertical wall. The speed of the ball is \(u\) before the impact and, as shown in Fig. 4.2, the impact turns the path of the ball through \(90 ^ { \circ }\). The coefficient of restitution in the collision between the ball and the wall is \(e\). Before the collision, the path is inclined at \(\alpha\) to the wall. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_294_590_1804_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Write down, in terms of \(u , e\) and \(\alpha\), the components of the velocity of the ball parallel and perpendicular to the wall before and after the impact.
   2. Show that \(\tan \alpha = \frac { 1 } { \sqrt { e } }\).
   3. Hence show that \(\alpha \geqslant 45 ^ { \circ }\).

1

1. Two small spheres, \(P\) of mass 2 kg and \(Q\) of mass 6 kg , are moving in the same straight line along a smooth, horizontal plane with the velocities shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_252_647_404_708} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Consider the direct collision of P and Q in the following two cases.
   1. The spheres coalesce on collision.
      (A) Calculate the common velocity of the spheres after the collision.
      (B) Calculate the energy lost in the collision.
   2. The spheres rebound with a coefficient of restitution of \(\frac { 2 } { 3 }\) in the collision.
      (A) Calculate the velocities of P and Q after the collision.
      (B) Calculate the impulse on P in the collision.
2. A small ball bounces off a smooth, horizontal plane. The ball hits the plane with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\arcsin \frac { 12 } { 13 }\) to it. The ball rebounds at an angle of \(\arcsin \frac { 3 } { 5 }\) to the plane, as shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_238_545_1695_767} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Calculate the speed with which the ball rebounds from the plane.
   Calculate also the coefficient of restitution in the impact.

1

1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
   1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
   2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
   3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .

7 Two small uniform smooth spheres A and B , of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere \(A\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is e .

1. Show that the velocity of B after the collision, in the original direction of motion of A , is \(\frac { 1 } { 5 } ( 1 + 6 e ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and find a similar expression for the velocity of A after the collision.
2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B .
   1. In the collision between A and B the spheres coalesce to form a combined body C . State the speed of C after the collision.
   2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of e .
   3. The total loss in kinetic energy due to the collision is 3 J . Determine the value of e.

2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\) 3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\)
   Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\)
   Explain the reasoning that they have used, showing any calculations that you make.
   4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
   Two of these classifications are deep water waves and shallow water waves.
   4
3. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
4. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)
   5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
   Find the maximum power output of the car.
   Fully justify your answer.

1 A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(v\).
2. Calculate the magnitude of the impulse on A when the tow-rope tightens.