A 6.5-kg bowling ball thrown with a speed of 5.4 m/s hits a stationary 1.75-kg pin. What is the velocity of each object? Assume that the coefficient of restitution is 0.75.
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To find the velocities of the bowling ball and the pin after the collision, we can use the principle of conservation of linear momentum and the coefficient of restitution.
Step 1: Calculate the initial momentum of the bowling ball and the pin.
The momentum (P) of an object is calculated by multiplying its mass (m) with its velocity (v).
For the bowling ball, the initial momentum (P1) can be calculated as:
P1 = mass x velocity = (6.5 kg) x (5.4 m/s) = 35.1 kg⋅m/s
For the pin, the initial momentum (P2) is zero since it is stationary.
Step 2: Calculate the total momentum before the collision.
The total momentum before the collision (Ptotal) is the sum of the individual momenta:
Ptotal = P1 + P2 = 35.1 kg⋅m/s
Step 3: Apply the coefficient of restitution to find the velocities after the collision.
The coefficient of restitution (e) is defined as the ratio of the relative velocity of separation (vs) to the relative velocity of approach (va) between the two objects after the collision:
e = vs / va
In this case, since the pin is initially stationary, va is the velocity of the bowling ball before the collision, and vs is the velocity of the bowling ball after the collision.
Step 4: Substitute the given values into the equation to solve for vs.
0.75 = vs / 5.4 m/s
vs = 0.75 x 5.4 m/s = 4.05 m/s
Step 5: Calculate the velocity of the pin after the collision.
Since the total momentum before and after the collision is the same, and the initial momentum of the pin is zero, the momentum of the pin after the collision (P2') is equal to the final momentum of the bowling ball (P1') after the collision.
P2' = P1' = m x v' (where v' is the final velocity of the bowling ball)
Since mass and final velocity are unknown, we'll use the principle of conservation of linear momentum to set up an equation:
P1 + P2 = P1' + P2'
35.1 kg⋅m/s + 0 = (6.5 kg) x v' + (1.75 kg) x v2'
v2' = 35.1 kg⋅m/s / 1.75 kg = 20 m/s
Therefore, after the collision:
- The bowling ball has a velocity of 4.05 m/s.
- The pin has a velocity of 20 m/s.
To find the velocities of the bowling ball and the pin after the collision, we can use the principles of conservation of linear momentum and the coefficient of restitution.
The conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Initially, the bowling ball is moving with a velocity of 5.4 m/s, and the pin is stationary, so the initial momentum of the system is:
Initial momentum before the collision = (mass of the bowling ball) x (velocity of the bowling ball) + (mass of the pin) x (velocity of the pin)
= (6.5 kg) x (5.4 m/s) + (1.75 kg) x (0 m/s)
The bowling ball hits the pin, and they both begin to move together after the collision. Let's denote the final velocity of the bowling ball and the pin as Vf.
According to the coefficient of restitution, the relative velocity of separation (after the collision) is proportional to the relative velocity of approach (before the collision). Mathematically, we can express this as:
Vf - 0 = -0.75 x (5.4 m/s - 0 m/s)
Simplifying the equation:
Vf = -0.75 x 5.4 m/s
Vf = -4.05 m/s
(Note: The negative sign indicates that the objects are moving in the opposite direction)
Now that we have the final velocity of the combined system, we can solve for the individual velocities of the bowling ball and the pin.
We can use the conservation of linear momentum again:
Mass of the bowling ball x velocity of the bowling ball (initial) + Mass of the pin x velocity of the pin (initial) = (Mass of the bowling ball + Mass of the pin) x (Velocity of the combined system)
(6.5 kg) x (5.4 m/s) + (1.75 kg) x (0 m/s) = (6.5 kg + 1.75 kg) x (-4.05 m/s)
Simplifying the equation:
35.1 kg * m/s = 35.75 kg * (-4.05 m/s)
Rearranging the equation to solve for the velocity of the pin:
Velocity of the pin (initial) = (35.75 kg * (-4.05 m/s) - 6.5 kg * (5.4 m/s)) / 1.75 kg
Velocity of the pin (initial) = -19.2829 m/s
Therefore, the velocity of the pin after the collision is approximately -19.28 m/s, opposite to the direction of the bowling ball.
To find the velocity of the bowling ball after the collision, we subtract the velocity of the pin:
Velocity of the bowling ball (final) = Velocity of the combined system - Velocity of the pin (initial)
Velocity of the bowling ball (final) = -4.05 m/s - (-19.2829 m/s)
Velocity of the bowling ball (final) = 15.2329 m/s
Therefore, the velocity of the bowling ball after the collision is approximately 15.23 m/s in the opposite direction as the initial motion.