A stationary billiard ball, mass 0.17 kg, is struck by an identical ball moving at 4.8 m/s. After the collision, the second ball moves off at 60° to the left of its original direction. The stationary ball moves off at 30° to the right of the moving ball's original direction. What is the final speed of initially moving ball
To find the final speed of the initially moving ball, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Let's denote the mass of each ball as m (mass of the balls = 0.17 kg) and the velocity of the initially moving ball as v (velocity of the ball = 4.8 m/s).
Before the collision:
Momentum of the initially moving ball = mv
Momentum of the stationary ball = 0 (since it is stationary)
After the collision, the second ball moves off at 60° to the left of its original direction. We can resolve this velocity into two components: one along the original direction (parallel to the initial velocity) and one perpendicular to the original direction.
Let's denote the final velocity of the initially moving ball as vf1 and the final velocity of the stationary ball as vf2.
Using conservation of momentum:
mv = mvf1 + 0.17 kg * vf2
Next, we need to find the components of the final velocities vf1 and vf2. The second ball moves off at 60° to the left of its initial direction, so its velocity can be broken down into two components:
vf1 = vf1_parallel + vf1_perpendicular
The stationary ball moves off at 30° to the right of the moving ball's original direction, so its velocity can also be broken down into two components:
vf2 = vf2_parallel + vf2_perpendicular
Since we know the angles (60° and 30°) and the initial velocity (4.8 m/s), we can find the components using trigonometry:
vf1_parallel = vf1 * cos(60°)
vf1_perpendicular = vf1 * sin(60°)
vf2_parallel = vf2 * cos(30°)
vf2_perpendicular = vf2 * sin(30°)
Now we can substitute these components back into the conservation of momentum equation:
mv = (vf1_parallel + vf2_parallel) + (vf1_perpendicular + vf2_perpendicular)
We also know that the mass of the balls (m) is equal to 0.17 kg. Substituting all the known values into the equation, we can solve for vf1.
0.17 kg * 4.8 m/s = [(vf1 * cos(60°)) + (vf2 * cos(30°))] + [(vf1 * sin(60°)) + (vf2 * sin(30°))]
Simplifying and rearranging the equation will allow us to solve for vf1, which will give us the final speed of the initially moving ball.