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Nick and the Red Pool Ball
Physics! Help!?
A red billiard ball moving at 4 m/s hits a green one of equal mass which is at rest. If the collision is perfectly elastic, what are the speeds of the red and the green ball after the collision?
After the collision, the red ball has zero velocity, while the green ball has a velocity of 4 m/s, in the same direction as the red ball was originally moving.
Let u_r and u_g be the velocities of the red and green balls before the collision, and let v_r and v_g be the velocities after the collision. Both balls have mass m.
Then, in general, conservation of momentum says that:
m*(u_r + u_g) = m*(v_r + v_g)
But the green ball was initially stationary (in the frame of reference of the observer), so:
u_r = v_r + v_g
or
v_r = u_r - v_g
(v_r)^2 = (u_r - v_g)^2 = (u_r)^2 - 2*u_r*v_g + (v_g)^2
In general, conservation of energy requires that:
0.5*m*(u_r^2 + u_g^2) = 0.5*m*(v_r^2 + v_g^2)
again simplifying this for the case of u_g = 0, we get:
(u_r)^2 = (v_r)^2 + (v_g)^2.
Plugging the expression for (v_r)^2 we obtained from the momentum-conservation equation into the above equation, we get:
(u_r)^2 = (u_r)^2 - 2*u_r*v_g + (v_g)^2 + (v_g)^2.
u_r*v_g - (v_g)^2 = 0
v_g * (u_r - v_g) = 0
This has two solutions, v_g = 0 and v_g = u_r. The first solution is the "trivial solution" and implies (from momentum or energy conservation) that there is no interaction between the balls; the red ball continues on with the same velocity after the collision, which is impossible geometrically (the green ball is in the way, so the red ball can't continue moving as it did before the collision). The second solution corresponds to the "real" situation, and says that the green ball has the same velocity as the red ball did before the collision, and from either momentum or energy conservation, the red ball must therefore have zero velocity after the collision.
See source.



























































