A 3.12 kg bowling ball traveling at 15.3 m/s to the right collides with a 2.15 kg bocce ball traveling at 7.52 m/s to the right. The final velocity of the bowling ball is 2.33 m/s to the left.
Determine the final velocity of the bocce ball. What is the magnitude of the force experienced by the balls if the collision lasts 533 ms? Was the collision elastic?
Apply conservation of momentum in one direction to get the final velocity of the bocce ball.
Divide the momentum change magnitude of either ball by 0.533 to get the average force magnitude.
Add up initial and final total kinetic energies. If they are equal, the collision was elastic.
To determine the final velocity of the bocce ball, we can use the principle of conservation of momentum. In a collision, the total momentum of the system before the collision is equal to the total momentum of the system after the collision.
The momentum (p) of an object is defined as the product of its mass (m) and velocity (v), so we can write the equation for conservation of momentum as:
Initial momentum of the system = Final momentum of the system
(m1 * v1) + (m2 * v2) = (m1 * vf1) + (m2 * vf2)
Where:
m1 = mass of the bowling ball = 3.12 kg
v1 = initial velocity of the bowling ball = 15.3 m/s to the right
m2 = mass of the bocce ball = 2.15 kg
v2 = initial velocity of the bocce ball = 7.52 m/s to the right
vf1 = final velocity of the bowling ball = 2.33 m/s to the left (negative velocity)
vf2 = final velocity of the bocce ball (unknown)
Using the above equation, we can solve for vf2:
(3.12 kg * 15.3 m/s) + (2.15 kg * 7.52 m/s) = (3.12 kg * 2.33 m/s) + (2.15 kg * vf2)
This equation can be simplified to:
47.856 + 16.208 = 7.2736 + (2.15 kg * vf2)
64.064 = 7.2736 + (2.15 kg * vf2)
Rearranging the equation, we get:
2.15 kg * vf2 = 64.064 - 7.2736
vf2 = (64.064 - 7.2736) / 2.15 kg
vf2 = 56.7904 / 2.15 kg
vf2 ≈ 26.422 m/s to the right
Therefore, the final velocity of the bocce ball is approximately 26.422 m/s to the right.
To calculate the magnitude of the force experienced by the balls, we can use Newton's second law of motion, which states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a):
F = m * a
The acceleration can be calculated using the change in velocity and the time taken during the collision:
a = (vf - vi) / t
Where:
vf = final velocity = 2.33 m/s to the left (negative velocity)
vi = initial velocity = 15.3 m/s to the right
t = time taken during the collision = 533 ms = 0.533 s
Plugging in the values:
a = (2.33 m/s - 15.3 m/s) / 0.533 s
a = -12.97 m/s / 0.533 s
a ≈ -24.34 m/s²
Using Newton's second law, we can now calculate the force:
F = m * a
For the bowling ball:
m1 = 3.12 kg
F1 = 3.12 kg * -24.34 m/s²
F1 ≈ -75.8688 N
For the bocce ball:
m2 = 2.15 kg
F2 = 2.15 kg * -24.34 m/s²
F2 ≈ -52.329 N
The magnitude of the force experienced by the balls during the collision is approximately 75.8688 N for the bowling ball and 52.329 N for the bocce ball.
To determine if the collision was elastic or not, we can calculate the coefficient of restitution (e). The coefficient of restitution measures the elasticity of a collision and is equal to the ratio of the relative velocity after the collision to the relative velocity before the collision.
e = (vf2 - vf1) / (v2 - v1)
v2 = initial velocity of the bocce ball = 7.52 m/s to the right
v1 = initial velocity of the bowling ball = 15.3 m/s to the right
vf1 = final velocity of the bowling ball = 2.33 m/s to the left (negative velocity)
vf2 = final velocity of the bocce ball = 26.422 m/s to the right
e = (26.422 m/s - 2.33 m/s) / (7.52 m/s - 15.3 m/s)
e = 24.092 m/s / -7.78 m/s
e ≈ -3.100
The coefficient of restitution, e, is approximately -3.100.
In elastic collisions, the coefficient of restitution is always positive (e > 0), indicating that the relative velocities after the collision are in the opposite direction as before the collision. However, in this case, the negative value of e (-3.100) indicates an inelastic collision.
Therefore, the collision between the bowling ball and the bocce ball was inelastic.