A billiard ball of mass of 0.879 kg and speed
0.602 m/s makes a head-on elastic collision
with an identical billiard ball at rest.
What is the kinetic energy of the second
ball after the collision?
What is the magnitude of its linear momentum?
To calculate the kinetic energy of the second ball after the collision, we first need to determine the velocity of the first ball after the collision using the principle of conservation of linear momentum. The law of conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Step 1: Find the initial momentum of the first ball.
Momentum (p) is calculated as the product of mass (m) and velocity (v). The initial momentum of the first ball is given by:
Initial momentum = mass x velocity
P1 = m1 x v1
Substituting the given values:
m1 = 0.879 kg (mass of the first ball)
v1 = 0.602 m/s (speed of the first ball before the collision)
P1 = 0.879 kg x 0.602 m/s
Step 2: Since the collision is head-on, the total momentum before the collision is equal to zero (as the second ball is at rest initially). This means that the initial momentum of the first ball is equal to the final momentum of both balls after the collision.
Step 3: Find the final momentum of both balls after the collision.
Suppose the final velocity of the first ball is v1' and the final velocity of the second ball is v2'. The final momentum of the first ball is given by:
Final momentum of ball 1 = mass of ball 1 x final velocity of ball 1
P1' = m1 x v1'
Since the second ball was initially at rest, its initial momentum was zero. Therefore, the final momentum of the second ball is given by:
Final momentum of ball 2 = mass of ball 2 x final velocity of ball 2
P2' = m2 x v2'
Since the balls undergo an elastic collision, the law of conservation of linear momentum can be expressed as:
Initial momentum of ball 1 = Final momentum of ball 1 + Final momentum of ball 2
P1 = P1' + P2'
Step 4: Rearrange and solve for the final velocity of the first ball, v1'.
v1' = (P1 - P2') / m1
Step 5: Calculate the initial momentum of the second ball.
The initial momentum of the second ball is zero since it was initially at rest.
Step 6: Calculate the final momentum of the second ball.
Final momentum of ball 2 = mass of ball 2 x final velocity of ball 2
P2' = m2 x v2'
Step 7: Calculate the kinetic energy of the second ball after the collision.
The kinetic energy of an object is given by 1/2 times the mass times the square of its velocity.
Kinetic energy of the second ball = (1/2) x m2 x (v2')^2
Step 8: Calculate the magnitude of linear momentum of the second ball.
Magnitude of linear momentum = mass x velocity
Magnitude of linear momentum of the second ball = m2 x v2'
Substitute the known values into the equations and calculate the final velocity, kinetic energy, and magnitude of linear momentum of the second ball after the collision.