2.9 Thermodynamics and Inelastic Collisions: Conservation of Momentum
In the previous section, we explored momentum and the two types of collisions: elastic and inelastic. Here, we dive deeper into inelastic collisions, where kinetic energy is not conserved due to transformations into heat, sound, or deformation. Let’s get started! 
Inelastic Collisions
In an inelastic collision, the total momentum is conserved, but kinetic energy is not. This occurs because some kinetic energy transforms into other energy forms. A completely inelastic collision involves the objects sticking together after the collision.
Key Characteristics:
Momentum is conserved.
Kinetic energy is not conserved.
Objects often deform or stick together.
Example:
Two carts of the same mass lie on a frictionless table. The first cart moves toward the second, which is stationary. After the collision, the two carts stick together. To find the final speed:
Calculate initial momentum:
Calculate final momentum (combined system):
Since momentum is conserved, . Solve for .
Note: While molecular collisions are often slightly inelastic, we frequently assume elasticity for simplicity in modeling.
Key Differences Between Elastic and Inelastic Collisions:
| Characteristic | Elastic Collision | Inelastic Collision |
|---|---|---|
| Kinetic Energy Conservation | Conserved | Not Conserved |
| Momentum Conservation | Conserved | Conserved |
| Post-Collision Behavior | Objects rebound | Objects deform or stick together |
| Energy Transformation | None | Some KE transforms into heat, sound, etc. |
Example Problems
Example Problem #1
Scenario:
Two carts:
Cart 1: Mass = 5 kg, Velocity = 3 m/s (right).
Cart 2: Mass = 2 kg, Velocity = 0 m/s (stationary).
After the collision:
Cart 1: Velocity = -1 m/s (left).
(a) Is this collision elastic or inelastic?
Initial KE:
Final KE:
Since , the collision is inelastic.
(b) What is the common final velocity?
Momentum Conservation:
(c) Initial and Final KE:
Initial KE: .
Final KE: .
Change in KE: .
Example Problem #2
Scenario:
Two bowling balls collide:
Ball 1: Mass = 20 kg, Velocity = 5 m/s (left).
Ball 2: Mass = 10 kg, Velocity = 0 m/s (stationary).
After collision:
Ball 1: Velocity = 2 m/s (right).
Ball 2: Velocity = 3 m/s (left).
(a) Is this collision elastic or inelastic?
Initial KE:
Final KE:
Since , the collision is inelastic.
(b) Change in KE:
Change in KE:
Conclusion
Inelastic collisions are essential to understanding real-world interactions, where energy transforms into other forms like heat or sound. While momentum is always conserved, recognizing the loss of kinetic energy helps classify collisions. Master these concepts to tackle thermodynamics and mechanics problems with confidence.
