Section 5.4: Elastic and Inelastic Collisions
Collisions can be categorized based on energy conservation:
- Elastic Collisions: Both momentum and kinetic energy are conserved.
- Inelastic Collisions: Momentum is conserved, kinetic energy is partially lost.
- Perfectly Inelastic: Objects stick together; maximum kinetic energy lost.
Elastic Collision Formulas (1D):
- \(v_{1f} = \frac{(m_1 - m_2)v_{1i} + 2 m_2 v_{2i}}{m_1 + m_2}\)
- \(v_{2f} = \frac{(m_2 - m_1)v_{2i} + 2 m_1 v_{1i}}{m_1 + m_2}\)
Example 1
A 2 kg ball moving at 3 m/s collides elastically with a 3 kg ball at rest. Find final velocities.
\(v_{1f} = \frac{(2-3)*3 + 2*3*0}{2+3} = -0.6 \text{ m/s}\)
\(v_{2f} = \frac{(3-2)*0 + 2*2*3}{2+3} = 2.4 \text{ m/s}\)
Example 2
Two cars collide inelastically and stick together. Car 1: 1000 kg, 20 m/s. Car 2: 1500 kg, 0 m/s. Find final velocity.
Total initial momentum: \( p_i = 1000*20 + 1500*0 = 20000 \text{ kg·m/s} \)
Total mass after collision: \( 1000 + 1500 = 2500 \text{ kg} \)
Final velocity: \( v_f = 20000 / 2500 = 8 \text{ m/s} \)
Practice Problems
- Two 4 kg carts collide elastically: velocities 6 m/s and -2 m/s. Find final velocities.
- A 0.5 kg ball at 5 m/s hits 0.5 kg ball at rest elastically. Find final velocities.
- Two skaters, 45 kg and 55 kg, push off each other and move in opposite directions. Find velocities.
- A 2 kg ball at 4 m/s collides inelastically with 3 kg ball at rest. Find final velocity.
- Perfectly inelastic collision: 1 kg and 2 kg moving 3 m/s and 0 m/s. Find final velocity.
- Elastic collision: 3 kg and 2 kg moving 5 m/s and 0 m/s. Find final velocities.
- A 1 kg puck hits 2 kg puck elastically. Velocities: 3 m/s and 0 m/s. Find post-collision velocities.
- Two carts, 3 kg each, collide elastically at 2 m/s and -1 m/s. Find final velocities.
- Perfectly inelastic collision: 0.2 kg and 0.3 kg balls moving 10 m/s and 0 m/s. Find final velocity.
- Two cars, 1000 kg and 1200 kg, collide inelastically. Car 1: 15 m/s, Car 2: 0 m/s. Find final velocity.