Oblique collision, find velocities/angles

5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the coefficient of restitution between the spheres.
3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
4. Find the mass of \(B\).

7 Two snooker balls, one white and one red, have equal mass. The balls are on a horizontal table \(A B C D\) The white ball is struck so that it moves at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to \(A B\) The white ball hits a stationary red ball.
After the collision, the white ball moves at a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(30 ^ { \circ }\) to \(A B\) After the collision, the red ball moves at a speed \(v \mathrm {~ms} ^ { - 1 }\) and at an angle \(\theta\) to \(C D\) When the collision takes place, the white ball is the same distance from \(A B\) as the distance the red ball is from CD The diagram below shows the table and the velocities of the balls after the collision. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-08_595_1370_1121_335} Not to scale After the collision, the white ball hits \(A B\) and the red ball hits \(C D\) Model the balls as particles that do not experience any air resistance.
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1. Explain why the two balls hit the sides of the table at the same time.
   7
2. Show that \(\theta = 17.0 ^ { \circ }\) correct to one decimal place.
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3. \(\quad\) Find \(v\) 7
4. Determine which ball travels the greater distance after the collision and before hitting the side of the table. Fully justify your answer.
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5. State one possible refinement to the model that you have used. \(8 \quad\) In this question use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) A rope is used to pull a crate, of mass 60 kg , along a rough horizontal surface.
   The coefficient of friction between the crate and the surface is 0.4 The crate is at rest when the rope starts to pull on it.
   The tension in the rope is 240 N and the rope makes an angle of \(30 ^ { \circ }\) to the horizontal.
   When the crate has moved 5 metres, the rope becomes detached from the crate.

6 Two smooth spheres, \(A\) and \(B\), have masses \(m\) and \(2 m\) respectively and equal radii. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is projected with speed \(u\) along the floor in a direction parallel to a smooth vertical wall and strikes \(B\) obliquely. Subsequently \(B\) strikes the wall at an angle \(\alpha\) with the wall. The coefficient of restitution between \(A\) and \(B\) and between \(B\) and the wall is 0.5. After \(B\) has struck the wall, \(A\) and \(B\) are moving parallel to each other.

1. Write down a momentum equation and a restitution equation along the line of centres for the impact between \(A\) and \(B\). Hence find the components of velocity of \(A\) and \(B\) in this direction after this first impact.
2. Find the value of \(\alpha\), giving your answer in degrees.

12 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_205_200_264_497} \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_284_899_349_753} A white snooker ball of mass \(m\) moves with speed \(u\) towards a stationary black snooker ball of the same mass and radius. Taking the \(x\)-axis to be the line of centres of the two balls at the moment of collision, the direction of motion of the white ball before the collision makes an angle of \(30 ^ { \circ }\) with the positive \(x\)-axis (see diagram).

1. Given that the coefficient of restitution is 0.9 , find the angle made with the \(x\)-axis by the velocity of the white ball after the collision.
2. Show that after the collision the white ball cannot have a negative \(x\)-component of velocity whatever the value of the coefficient of restitution.

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(m\) kg respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a\text{ ms}^{-1}\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b\text{ ms}^{-1}\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2\text{ ms}^{-1}\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2\text{ ms}^{-1}\) in a direction making an angle of \(45°\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{3}\).

1. Show that \(a\cos \alpha = \frac{5}{3}\sqrt{2}\) and find \(b\cos \beta\). [7]
2. Find the values of \(a\) and \(\alpha\). [4]

\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(3m\) kg respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5\) m s\(^{-1}\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac{4}{5}\), and \(B\) is moving with speed \(3\frac{1}{4}\) m s\(^{-1}\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac{5}{13}\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{5}\).

1. Find the speed of \(A\) after the collision. Find also the component of the velocity of \(B\) along the line of centres after the collision. [7]

\(B\) subsequently hits the wall.

1. Explain why \(A\) and \(B\) will have a second collision if the coefficient of restitution between \(B\) and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. [3]

Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is \(5\) kg and the mass of \(B\) is \(3\) kg. At the instant before they collide, • the velocity of \(A\) is \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} The coefficient of restitution for collisions between the two discs is \(\frac{3}{4}\). Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. [7]