Final answer:
To calculate the speed of two carts after a spring between them is released, conservation of energy is used, considering the potential energy of the spring and the kinetic energy of each cart. The speed of each 2 kg cart, after the spring with stiffness 500 N/m decompresses by 1 cm, is found to be approximately 0.112 m/s.
Step-by-step explanation:
Calculating the Speed of Two Carts After Spring Release
When the spring between two carts decompresses, it transfers potential energy into kinetic energy equally to both carts. To find the speed of each cart after the spring is released, we apply conservation of energy. The initial potential energy stored in the spring is given by U = 1/2 k x^2, where k is the stiffness of the spring and x is the compression distance. Since the total energy is conserved, this potential energy is converted into kinetic energy of both carts. The kinetic energy of each cart is given by K = 1/2 m v^2, where m is the mass of the cart and v is the speed of the cart.
Given:
- Mass of each cart (m) = 2 kg
- Stiffness of the spring (k) = 5 J/cm^2 = 500 N/m (since 1 J/cm^2 = 100 N/m)
- Compression distance (x) = 1 cm = 0.01 m
First, calculate the potential energy stored in the spring:
- U = 1/2 k x^2
- U = 1/2 (500 N/m) (0.01 m)^2
- U = 0.025 J
Then, this energy is converted into the kinetic energy of both carts, so the kinetic energy of each cart is half of that:
- K = U / 2
- K = 0.025 J / 2
- K = 0.0125 J
Solving for the speed (v) of each cart using the kinetic energy expression:
- K = 1/2 m v^2
- 0.0125 J = 1/2 (2 kg) v^2
- 0.0125 J = (1 kg) v^2
Solving for v gives:
- v^2 = 0.0125 m^2/s^2
- v = √(0.0125 m^2/s^2)
- v = 0.112 m/s
Thus, the speed of each cart after the spring is released is approximately 0.112 m/s.