A white billiard ball with mass m w = 1.39 kg is moving directly to the right wi

Question

A white billiard ball with mass mw = 1.39 kg is moving directly to the right with a speed of v = 3.24 m/s and collides elastically with a black billiard ball with the same mass mb = 1.39 kg that is initially at rest. The two collide elastically and the white ball ends up moving at an angle above the horizontal of w = 67° and the black ball ends up moving at an angle below the horizontal of b = 23°.

1-What is the final speed of the white ball?

2-

What is the final speed of the black ball?

3-

What is the magnitude of the final total momentum of the system?

4-

What is the final total energy of the system?

Explanation / Answer

Let horizontal direction be x direction
initial momentum in x direction = 1.39*3.24 = 4.5036 Kgm/s
initial momentum in y direction = 0 Kgm/s
let the final speed of while ball be v1 and that of black ball be v2
final momentum in x direction = 1.39*v1*cos 67 + 1.39*v2*cos 23
final momentum in y direction = 1.39*v1*sin 67 - 1.39*v2*sin 23

use conservation of momentum in y direction:
initial momentum in y direction = final momentum in y direction
0 = 1.39*v1*sin 67 - 1.39*v2*sin 23
0 = 1.28*v1 - 0.54*v2
v2 = 2.37*v1

use conservation of momentum in x direction:
initial momentum in x direction = final momentum in x direction
4.5036 = 1.39*v1*cos 67 + 1.39*v2*cos 23
4.5036 = 0.54*v1*cos 67 + 1.28*v2
put v2 = 2.37*v1
4.5036 = 0.54*v1 + 1.28* 2.37*v1
4.5036 = 0.54*v1 + 3.03*v1
v1 = 1.26 m/s
Answer: 1.26 m/s

2)
v2 = 2.37*v1
= 2.37*1.26
= 2.99 m/s
Answer: 2.99 m/s

3)
since momentum is conserved in elastic collision,
total final momentum = total initial momentum
= 4.5036 kgm/s
Answer: 4.5036 kgm/s

4)
total final energy = 0.5*m1*v1^2 +0.5*m2*v2^2
= 0.5*1.39*1.26^2 + 0.5*1.39*2.99^2
= 7.32 J
Answer: 7.32 J

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