Two-sphere oblique collision

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\includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-08_561_1068_255_500} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision A's direction of motion makes an angle of \(\alpha ^ { \circ }\) with the line of centres, and \(B\) 's direction of motion is perpendicular to that of \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\). Immediately after the collision, \(B\) moves in a direction at right angles to the line of centres.

1. Show that \(\tan \alpha = \frac { 1 + e } { 1 - e }\).
2. Given that \(\tan \alpha = 2\), find the speed of \(A\) after the collision.

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\includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-10_339_983_258_541} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after the collision is \(\frac { 4 \mathrm { u } \cos \theta } { 3 ( 1 + \mathrm { k } ) }\).
   70\% of the total kinetic energy of the spheres is lost as a result of the collision.
2. Given that \(\tan \theta = \frac { 1 } { 3 }\), find the value of \(k\).

6 Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) after the collision is \(\frac { 3 \mathrm { u } \cos \alpha } { 2 ( 1 + \mathrm { k } ) }\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k , u\) and \(\alpha\).
   After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
2. Given that \(\tan \alpha = \frac { 2 } { 3 }\), find the possible values of \(k\).

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\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-02_358_1019_287_523} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2 m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac { 1 } { 2 } u\) respectively. Immediately before the collision, \(A\) 's direction of motion is along the line of centres, and \(B\) 's direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac { 1 } { 4 } u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-02_2716_37_141_2012}