A 0.34-kg cart and a 0.18-kg cart are held together with a compressed spring between them. When they are released, the 0.34-kg cart moves at 1.1 m/s to the right.
Part A
How much elastic potential energy was stored in the spring before the release?
To calculate the elastic potential energy stored in the spring, we need to use the formula:
Elastic Potential Energy = 0.5 * k * x^2
Where:
- k is the spring constant
- x is the displacement of the spring from its equilibrium position
Since the carts are held together with a compressed spring, we can assume that the displacement x is the distance that the 0.34-kg cart moved to the right after the release.
Given:
- Mass of the 0.34-kg cart (m1) = 0.34 kg
- Mass of the 0.18-kg cart (m2) = 0.18 kg
- Velocity of the 0.34-kg cart (v1) = 1.1 m/s
First, we need to find the velocity of the combined system after the release. Since the two carts are held together, they will move with the same velocity. We can use the law of conservation of momentum to find this velocity.
Momentum before release = Momentum after release
(m1 * v1) + (m2 * 0) = (m1 + m2) * v
Where:
- v is the velocity of the combined system after release
Rearranging the equation, we get:
v = (m1 * v1) / (m1 + m2)
Next, we can calculate the kinetic energy of the system using the formula:
Kinetic Energy = 0.5 * (m1 + m2) * v^2
Substituting the values we have, we can calculate the kinetic energy of the system.
Finally, since the system started with zero potential energy, the total mechanical energy of the system is equal to the kinetic energy of the system. Therefore, the elastic potential energy stored in the spring before release is equal to the calculated kinetic energy of the system.