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the red ball comes to rest its vector change is solely of magnitude.

That is all there is to see. A diagram that illustrates Newton?s equal and opposite force and momentum change. X is the difference between the orthodox view and reality.

In Snooker it is called a kiss, when both balls are moving. Do try it. Roll a blue ball down the table and hit it with a cue ball at right angles to its line of motion. JFD, I have and, with a bit of patience, you will hit the sweet spot.

I need help here. Please show me the correct mathematics.

This is a discovery I made many years ago.

The diagram shows a blue ball traveling at a velocity of A-C

which is struck by a red ball traveling at a velocity of B-C.

The Blue ball is deflected with a resultant velocity C-D, the vector change is solely of direction, the red ball comes to rest its vector change is solely of magnitude.

There is not enough information given to have any idea where the balls would end up. We do not know the speed of either ball nor do we know at what angle the balls hit each other.

My mistake. I made the assumption that vector diagrams were commonplace and that anyone who would be contributing to the thread would recognise that at a velocity of A-C was referencing a vector diagram.

The diagram is drawn to scale. AC is 4 units long, BC is 3 units long, and therefore CX is 5 units long the vector sum of AC, BC

I did this experiment in school, and your assertions have been falsified.

That change is shown in the diagram as the red mass coming to rest, no residual momentum, the blue mass moves on the line of CD an unchanged speed of 4 units and also along the line of BC at speed of 3 units.

bringing red to rest, and changing the direction of blue, but crucially not the speed.

In the first law, an object will not change its motion unless a force acts on it.

I can't make any sense of the diagram supplied

You just said the speed did change! The blue ball was moving originally at 4 units in the CD (x) direction. After the collision it is moving those 4 units plus 3 units in the BC (y) direction. That is an increase in speed.

The momentum of the blue ball is changed by its change in direction, not by a change in speed.

Why does the blue ball not push the red ball right to left?How does it know that it's only allowed to push it up (to arrest that component of its motion) but not from right to left?

If you hit the blue ball centrally, it neither slows down nor speeds, up