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 can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum: In an isolated system, the total momentum before a collision is equal to the total momentum after the collision.
Let's assume the initial velocity of the second ball (the stationary ball) is v2, and the final velocity of the second ball is vf2.
Before the collision:
Momentum of ball 1 (m1) = m1 * v1 (where v1 = 5.80 m/s)
Momentum of ball 2 (m2) = m2 * v2 (where v2 = 0 m/s)
After the collision:
Momentum of ball 1 (m1) = m1 * vf1 (where vf1 = 4.81 m/s)
Momentum of ball 2 (m2) = m2 * vf2
Using the conservation of momentum, we can write the equation:
m1 * v1 + m2 * v2 = m1 * vf1 + m2 * vf2
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.
Before the collision:
Kinetic energy of ball 1 = 0.5 * m1 * (v1)^2
Kinetic energy of ball 2 = 0.5 * m2 * (v2)^2
After the collision:
Kinetic energy of ball 1 = 0.5 * m1 * (vf1)^2
Kinetic energy of ball 2 = 0.5 * m2 * (vf2)^2
Using the conservation of kinetic energy, we can write the equation:
0.5 * m1 * (v1)^2 + 0.5 * m2 * (v2)^2 = 0.5 * m1 * (vf1)^2 + 0.5 * m2 * (vf2)^2
Now we have two equations (momentum and kinetic energy) and two unknowns (vf2 and vf2) that we can solve simultaneously.
Solving the equations:
From the momentum equation, we can write:
m1 * v1 + m2 * v2 = m1 * vf1 + m2 * vf2 (Equation 1)
From the kinetic energy equation, we can write:
0.5 * m1 * (v1)^2 + 0.5 * m2 * (v2)^2 = 0.5 * m1 * (vf1)^2 + 0.5 * m2 * (vf2)^2 (Equation 2)
We also have the angle between the initial and final velocities of ball 1 (34.0°). This angle can be used to find the x and y components of vf1:
vf1x = vf1 * cos(34.0°) (horizontal component)
vf1y = vf1 * sin(34.0°) (vertical component)
Now we can substitute vf1x and vf1y into Equation 1 to solve for vf2.
With vf2 determined, we can then find the direction of the struck ball's velocity by calculating the angle it makes with respect to the original line of motion.
To summarize, we need to solve the simultaneous equations of conservation of momentum and conservation of kinetic energy and use the given angle to find the x and y components of vf1. Then we can plug those values into the momentum equation to solve for vf2. Finally, we calculate the angle of vf2 with respect to the original line of motion.