A cue ball initially moving at 4.3m/s strikes a stationary eight ball of the same size and mass. After the collision, the cue ball’s final speed is 2.3 m/s at angle 1 with respect to it's original line of motion. The 8 ball moves with unknown speed at an angle 2 unknown with respect to its original line of motion below the horizontal.
Find the cue ball’s angle 1 with respect to its original line of motion above the horizontal. Consider this to be an elastic collision (ignoring friction and rotational motion).
Two 2.5 kg masses are 2.5 m apart on a frictionless table. Each has +1.5 µC of charge.
To solve this problem, we will use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
In an elastic collision between two objects, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. In this case, the cue ball and the eight ball have the same mass.
Let's denote the mass of the cue ball as m and the speed before the collision as v1.
The initial momentum of the system is given by:
Initial momentum = momentum of the cue ball + momentum of the eight ball
= m * v1 + 0 (as the eight ball is stationary)
After the collision, the cue ball has a final speed of v2 and the eight ball has a final speed of v3. We need to find the angle of the cue ball's final velocity vector with respect to its original line of motion.
The final momentum of the system is given by:
Final momentum = momentum of cue ball after collision + momentum of eight ball after collision
= m * v2 * cos(angle1) + m * v3 * cos(angle2)
Since the total momentum is conserved, we can equate the initial and final momenta to find the relationship between v1, v2, v3, and the angles:
m * v1 = m * v2 * cos(angle1) + m * v3 * cos(angle2) --(1)
2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of an object is given by half the product of its mass and the square of its velocity.
The initial kinetic energy of the system is given by:
Initial kinetic energy = kinetic energy of the cue ball + kinetic energy of the eight ball
= 0.5 * m * v1^2 + 0 (as the eight ball is stationary)
After the collision, the final kinetic energy of the system is given by:
Final kinetic energy = kinetic energy of cue ball after collision + kinetic energy of eight ball after collision
= 0.5 * m * v2^2 + 0.5 * m * v3^2
Since the total kinetic energy is conserved, we can equate the initial and final kinetic energies to find the relationship between v1, v2, and v3:
0.5 * m * v1^2 = 0.5 * m * v2^2 + 0.5 * m * v3^2 --(2)
Now, we have two equations (equation 1 and equation 2) with two unknowns (angle1 and angle2). To solve for angle1, we can rearrange equation 1 to isolate cos(angle1):
v1 = v2 * cos(angle1) + v3 * cos(angle2) --(3)
Taking the ratio of equation 1 and equation 2, we can eliminate the mass and solve for cos(angle1):
(v1^2) / (v1^2) = (v2 * cos(angle1) + v3 * cos(angle2)) / (v2^2 + v3^2)
1 = cos(angle1) * (v2 / v1) + cos(angle2) * (v3 / v1)
Rearranging the equation, we get:
cos(angle1) = (1 - cos(angle2) * (v3 / v1)) / (v2 / v1)
Finally, taking the inverse cosine of both sides:
angle1 = arccos((1 - cos(angle2) * (v3 / v1)) / (v2 / v1))
Now you can plug in the given values for v1, v2, v3, and angle2, and calculate angle1.