Two billiard balls of identical mass move toward each other as shown in the figu
ID: 2206222 • Letter: T
Question
Two billiard balls of identical mass move toward each other as shown in the figure. Assume that the collision between them is perfectly elastic. If the initial velocities of the balls are +32.5 cm/s and -20.3 cm/s, what is the velocity of each ball after the collision? Assume friction and rotation are unimportant. I found the v1f=-20.3cm/s and the v2f=32.5cm/s, both of these answers were correct. But now I am being asked to adjust the masses. I am not sure if I am doing this correctly.
Find the final velocity of the two balls if the ball with velocity v2i = -20.3 cm/s has a mass equal to half that of the ball with initial velocity v1i = +32.5 cm/s.
Explanation / Answer
The answer is simply -20.9 cm/s and +28.3 cm/s to the opposite direction. (the ball moving -20.9 cm/s will go +28.3 cm/s and vice versa) For perfectly elastic ideal(no friction or loss of energy through any other means) collision, the momentum of the system before is equal the momentum of the system after. By system, we mean the two billiard balls. So we have the ff. equations: momentum = mass x velocity (p=mv) p of ball 1 before collision + p of ball 2 before collision = p of ball 1 after collision + p of ball 2 after collision. (conservation of momentum) KE = 1/2(m)(v)^2 KE before collision = KE after collision (conservation of energy) These are the given: (assuming + is to the right and - is to the left) v of ball 1 before collision = 28.3 cm/s (to the right) v of ball 2 before collision = 20.9 cm/s (to the left) m of ball 1 = 1/2m (whatever m is) [NOTE: mass is always constant] m of ball 2 = 1/2m Find: v of ball 1 after collision | let this be x (or any other variable) v of ball 2 after collision | let this be y (or any other variable) Plug in the values... 1/2(28.3) +1/2(-20.9) = 1/2x + 1/2y 28.3-20.9 = x+y 7.4 = x+y x = 7.4 - y 1/2(1/2)(28.3)^2 + 1/2(1/2)(-20.9)^2 = 1/2(1/2)x^2 + 1/2(1/2)y^2 28.3^2 + 20.9^2 = x^2 + y^2 800.89 + 436.81 = (7.4-y)^2 + y^2 = (54.76 - 14.8y + y^2) + y^2 1237.7 = 2y^2 - 14.8y + 54.76 0 = 2y^2 - 14.8y +54.76 - 1237.7 0 = 2(y^2 - 7.4y - 591.47) 0 = (y-28.3)(y+20.9) y=28.3 or y=-20.9, but since y is coming from the right, it has to return to the right (inelastic collision of 2 equal masses moving towards each other) so y=28.3 since x = 7.4-y, x = 7.4 - 28.3,