A billiard ball moving at 5.80 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.81 m/s at an angle of 34.0° with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball's velocity after the collision and its direction.
To solve this problem, we will apply the principles of conservation of momentum and conservation of kinetic energy in an elastic collision.
Let's first assign labels to the given information:
- Ball 1 (moving ball): mass = m1, initial velocity = v1, final velocity = v1', angle with original line of motion = θ1
- Ball 2 (stationary ball): mass = m2, final velocity = v2, angle with the original line of motion = θ2
Now, let's define the conservation equations for momentum and kinetic energy:
Conservation of momentum:
m1 * v1 = m1 * v1' * cos(θ1) + m2 * v2 * cos(θ2) --> Equation 1
Note: We consider only the x-component of velocities because there is no motion in the y-direction after the collision.
Conservation of kinetic energy:
(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2^2 --> Equation 2
Now, let's plug in the given values into the equations and solve for the unknowns.
Given:
m1 = m2 (both balls have the same mass)
v1 = 5.80 m/s
v1' = 4.81 m/s
θ1 = 34.0°
Equation 1:
m1 * v1 = m1 * v1' * cos(θ1) + m2 * v2 * cos(θ2)
m1 * 5.80 = m1 * 4.81 * cos(34.0°) + m2 * v2 * cos(θ2)
Equation 2:
(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2^2
(1/2) * m1 * 5.80^2 = (1/2) * m1 * 4.81^2 + (1/2) * m2 * v2^2
Since there are two unknowns (v2 and θ2) in these equations, we cannot solve them directly. However, by manipulating these equations, we can eliminate θ2 and express v2 in terms of the other variables.
From Equation 1, we can isolate v2 * cos(θ2):
m2 * v2 * cos(θ2) = m1 * 5.80 - m1 * 4.81 * cos(34.0°)
v2 * cos(θ2) = (m1 * 5.80 - m1 * 4.81 * cos(34.0°)) / m2 --> Equation 3
Substituting this expression into Equation 2, we can solve for v2:
(1/2) * m1 * 5.80^2 = (1/2) * m1 * 4.81^2 + (1/2) * m2 * ((m1 * 5.80 - m1 * 4.81 * cos(34.0°)) / m2)^2
Simplifying this equation:
m1 * 5.80^2 = m1 * 4.81^2 + ((m1 * 5.80 - m1 * 4.81 * cos(34.0°)) / m2)^2
Now, we have an equation with only one unknown, v2. We can solve this equation to find the velocity of the second ball after the collision.
Once we have found v2, we can substitute it back into Equation 3 to find θ2, the angle with respect to the original line of motion.