Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Pool balls colliding headon
Two pool balls collide headon, so that the collision is restricted to one dimension. Pool balls are constructed so as to lose as little kinetic energy as possible in a collision, so under the assumption that no kinetic energy is converted to any other form of energy, what can we predict about the results of such a collision? Pool balls have identical masses, so we use the same symbol m for both. Conservation of energy and no loss of kinetic energy give us the two equations
The masses and the factors of 1/2 can be divided out, and we eliminate the cumbersome subscripts by replacing the symbols v_{1i ,...} with the symbols A,B,C, and D: A + B = C + D A^{2} + B^{2} = C^{2} + D^{2} . A little experimentation with numbers shows that given values of A and B, it is impossible to find C and D that satisfy these equations unless C and D equal A and B, or C and D are the same as A and B but swapped around. A formal proof of this fact is given in the sidebar. In the special case where ball 2 is initially at rest, this tells us that ball 1 is stopped dead by the collision, and ball 2 heads off at the velocity originally possessed by ball 1. This behavior will be familiar to players of pool. Gory Details of the Proof Above The equation A + B = C + D says that the change in one ball's velocity is equal and opposite to the change in the other's. We invent a symbol x = C  A for the change in ball 1's velocity. The second equation can then be rewritten as A^{2}+B^{2} = (A+x)^{2}+(Bx)^{2}. Squaring out the quantities in parentheses and then simplifying, we get 0 = Ax  Bx + x^{2}. The equation has the trivial solution x = 0, i.e., neither ball's velocity is changed, but this is physically impossible because the balls cannot travel through each other like ghosts. Assuming x 6≠0, we can divide by x and solve for x = B  A. This means that ball 1 has gained an amount of velocity exactly sufficient to match ball 2's initial velocity, and viceversa. The balls must have swapped velocities.


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