A white billiard ball with mass mw = 1.32 kg is moving directly to the right with a speed of v = 2.91 m/s and collides elastically with a black billiard ball with the same mass mb = 1.32 kg that is initially at rest. The two collide elastically and the white ball ends up moving at an angle above the horizontal of ¦Èw = 20¡ã and the black ball ends up moving at an angle below the horizontal of ¦Èb = 70¡ã.
A cue ball rolling across a pool table at 5.7 m/s collides with a stationary eight ball of equal mass. The cue ball stops after the collision. What is the velocity of the eight ball after the collision?
To solve this problem, we need to use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
Before the collision, the momentum of both balls is zero as the black ball is at rest. After the collision, the total momentum of the system (both balls) remains the same.
Let's define the positive direction to the right. The momentum of the white ball before the collision is given by:
pw_initial = mw * vw_initial
where mw is the mass of the white ball (1.32 kg) and vw_initial is its initial velocity (2.91 m/s).
After the collision, the white ball moves at an angle of 20 degrees above the horizontal. Decomposing its velocity into horizontal (x) and vertical (y) components, we get:
vw_final_x = vw_final * cos(20°)
vw_final_y = vw_final * sin(20°)
Similarly, the black ball moves at an angle of 70 degrees below the horizontal. Its velocity components after the collision are:
vb_final_x = vb_final * cos(70°)
vb_final_y = vb_final * sin(70°)
The total momentum of the system after the collision is zero:
pw_final_x + pb_final_x = 0
pw_final_x = -pb_final_x
Now we can write the momentum equations for both components, using conservation of momentum:
mw * vw_initial = mw * vw_final_x + mb * vb_final_x
0 = mw * vw_final_y + mb * vb_final_y
2. Conservation of kinetic energy:
Since the collision is elastic, the total kinetic energy before and after the collision remains the same.
The initial kinetic energy is:
Ke_initial = (1/2) * mw * (vw_initial)^2
After the collision, the kinetic energy of the white ball is:
Ke_final_white = (1/2) * mw * (vw_final)^2
And the kinetic energy of the black ball is:
Ke_final_black = (1/2) * mb * (vb_final)^2
Using the conservation of kinetic energy, we have:
Ke_initial = Ke_final_white + Ke_final_black
Solving the equations for momentum and kinetic energy will give us the final velocities of the white and black balls.
Note: It is important to convert the angle values to radians before using any trigonometric functions.
You will have to solve this in two directions of momentum.
let the intial direction be x.
You know the intial momentum, and masses, and let the final be vwhite'x and vblack'x
the initial momentum in the y directtion is zero, so the sum of the final y momentums is zero, which leads to
vwhite'y=-vblack'y
With those two equations, and the angles, you can solve for the final velocities