A white billiard ball with mass mw = 1.5 kg is moving directly to the right with

Question

A white billiard ball with mass mw = 1.5 kg is moving directly to the right with a speed of v = 3.14 m/s and collides elastically with a black billiard ball with the same mass mb = 1.5 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 = 37° and the black ball ends up moving at an angle below the horizontal of b = 53°. 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 final velocity of white ball be vw and that of black ball be vb.
Initial momentum in x direction = mw*viw + mb *vib = 1.5*3.14 + 0 = 4.71 Kgm/s
Initial momentum in y direction = 0
final momentum in x direction = 1.5*vw*cos37 +1.5*vb*cos53
final momentum in y direction = 1.5*vw*sin 37 - 1.5*vb*sin 53

use conservation of momentum in y direction
final momentum in y direction =Initial momentum in y direction
1.5*vw*sin 37 - 1.5*vb*sin 53 = 0
vw = 1.33 vb

use conservation of momentum in x direction
final momentum in x direction =Initial momentum in x direction
1.5*vw*cos37 +1.5*vb*cos53 = 4.71
1.2 vw + 0.9 vb = 4.71
put: vw = 1.33 vb
1.2*1.33*vb + 0.9vb = 4.71
1.6vb+0.9vb = 4.71
vb=1.9 m/s
vw = 1.33*vb = 1.33*1.9 = 2.5 m/s
a)
Answer: 2.5 m/s
b)
Answer: 1.9 m/s
c)
final momentum in x direction = 1.5*vw*cos37 +1.5*vb*cos53
= 1.5*2.5*cos37 +1.5*1.9*cos53
= 4.71 Kgm/s
final momentum in y direction = 0
so magnitude of final momentum = 4.71 Kgm/s
d)
Finally they have kinetic energy
E = 0.5*mb*vb^2 + 0.5*mw*vw^2
=0.5*1.5*1.9^2 + 0.5*1.5*2.5^2
=7.4 J
Answer: 7.4 J

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