2.8 Thermodynamics and Elastic Collisions: Conservation of Momentum

2.8 Thermodynamics and Elastic Collisions: Conservation of Momentum

Types of Collisions

When the net external force acting on an object is zero, its momentum is conserved. While some forces in one direction may cancel out, other directions could have unbalanced forces. For instance, a ball in free fall might have zero net force in the x-direction, but the y-direction experiences a net gravitational force.

Collisions occur when two objects strike each other, either rebounding or sticking together. Despite momentum being conserved, kinetic energy may not always be conserved as it can transform into other energy forms like heat or sound.

Key Points:

• Momentum is conserved in all types of collisions.

• Kinetic energy is conserved only in elastic collisions.

• In inelastic collisions, some kinetic energy is transformed into other forms of energy.

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. For problem-solving, use the following approaches:

1. Conservation of Momentum:

    

   Where:

   • : Initial momentum

   • : Final momentum

2. Conservation of Kinetic Energy:

    

   Where:

   • : Initial kinetic energy

   • : Final kinetic energy

For two-dimensional problems, consider both x and y directions separately. If vector sums or angles are involved, use trigonometry to resolve components.

Example Problem

Scenario:

• A 2 kg cart moves to the right at 3 m/s and collides with a stationary 1 kg cart.

• After the collision:

  • The 2 kg cart moves to the right at 2 m/s.

  • The 1 kg cart moves to the right at 1 m/s.

1. Classify the Collision:

Elastic or Inelastic?

• Initial Kinetic Energy:

• Final Kinetic Energy:

Since , the collision is elastic.

2. Justify the Use of Conservation Laws:

• Conservation of Momentum: Total system momentum remains unchanged.

• Conservation of Kinetic Energy: Valid for elastic collisions, allowing the calculation of unknowns.

3. Solve for Missing Variables:

Using conservation equations:

Substituting known values:

4. Validate Using Kinetic Energy:

Initial:

Final:

Both momentum and kinetic energy are conserved, confirming the solution.

Signs and Conventions

Momentum:

• Momentum direction determines sign.

• Solve separately for x and y components in 2D problems.

Kinetic Energy:

• Always positive since it depends on the square of velocity.

Conclusion

Elastic collisions are characterized by the conservation of both kinetic energy and momentum, while inelastic collisions conserve only momentum. These principles are foundational to solving problems in thermodynamics and mechanics.