Oblique collision of spheres

Two spheres collide at an angle (not head-on); resolve velocities along and perpendicular to line of centres, apply momentum and restitution in appropriate direction.

3 questions · Challenging +1.6

6.03c Momentum in 2D: vector form6.03d Conservation in 2D: vector momentum6.03k Newton's experimental law: direct impact
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CAIE Further Paper 3 2022 November Q6
8 marks Challenging +1.8
6 \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.
  1. Find the value of \(k\).
  2. Find the loss in the total kinetic energy as a result of the collision.
CAIE Further Paper 3 2022 November Q6
8 marks Challenging +1.8
6 \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.
  1. Find the value of \(k\).
  2. Find the loss in the total kinetic energy as a result of the collision.
CAIE Further Paper 3 2020 Specimen Q4
9 marks Challenging +1.2
\includegraphics{figure_4} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal surface. Sphere \(A\) is moving on the surface with speed \(u\) at an angle of \(30°\) to the line of centres of \(A\) and \(B\) when it collides with \(B\) (see diagram). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac{\sqrt{3}}{6}u(1 + e)\) and find the speed of \(A\) after the collision. [6]
  2. Given that \(e = \frac{1}{2}\), find the loss of kinetic energy as a result of the collision. [3]