A-level Mathematics/OCR/M3/Impulse and Momentum

Momentum in 2 dimensions
M2 builds on work done in M1, it doesn't introduce any new equations, however it introduces the use of vectors to describe Impulse, Momentum and the velocities involved.

defining coordinate systems
Now we are working in 2 dimensions it will simplify problems if we break the vectors involved down into their component parts, but first we must define the coordinates we are going to use.

A uniform sphere impacting a smooth surface(such as a snooker ball hitting the side of a table)
In these kind of questions where a sphere is hitting a solid unmovable surface it will be useful to use the direction parallel to the surface as the x direction, and the direction perpendicular to the surface to be the y direction. With this convention we can see that an impulse that occurs on impact will only act in the y direction, therefore the x component of the velocity of the impacting sphere will be the same before and after the impact, and the impulse I will equal M(Uy+Vy) (I = M(V-U) I have taken Uy to have a negative velocity, having the positive y axes going away from the surface.)

I will not derive it here but if we are given the angle of approach α and the angle of separation β of the sphere we can work out the coefficient of restitution e using:

e = tan(β)/tan(α)

Oblique impacts
This is similar to the above instance, however it is now two uniform spheres colliding. So instead of using the surface of the spheres as the basis of our coordinate system, we should use axes that are the same for both spheres at the time of impact, this is called the line of centres. Where the two spheres collide, the line of centres is the line connecting the centres of the two spheres, we will use this now as our x direction, which makes a change to our above assertion that the velocity in the x direction is unchanging in the impact. we have moved our axes round so that now it is the velocities of the spheres in the y direction that are unchanged in the impact.

therefore we have that Vy = Uy for both spheres.

Now in the x direction we make a return to M1 and 2, in the x direction we use the ideas of conservation of momentum, because as with the example of the ball and surface, the impulse must act along the line of centres, hence only changing the x components of the velocities of the spheres.