Section 4.5: Inelastic Collisions

In an inelastic collision, momentum is conserved, but kinetic energy is not. Colliding objects may stick together or deform, losing some kinetic energy as heat, sound, or deformation.

\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \] For a perfectly inelastic collision (objects stick together): \[ m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f \]

Here: \( m_1, m_2 \) are masses, \( v_{1i}, v_{2i} \) initial velocities, \( v_{1f}, v_{2f} \) final velocities, and \( v_f \) is the common velocity after sticking.

Example 1

A 2 kg cart moving at 3 m/s collides inelastically with a 3 kg cart at rest. Find their velocity after collision if they stick together.

Using momentum conservation:
\( (2)(3) + (3)(0) = (2+3) v_f \)
\( 6 = 5 v_f \Rightarrow v_f = 1.2 \, \text{m/s} \)

Example 2

Two ice skaters, 50 kg and 70 kg, move toward each other at 2 m/s and -1 m/s. They grab each other and move together. Find the final velocity.

Momentum before: \( (50)(2) + (70)(-1) = 100 - 70 = 30 \, \text{kg·m/s} \)
Total mass: 50 + 70 = 120 kg
Final velocity: \( v_f = 30 / 120 = 0.25 \, \text{m/s} \)

Practice Problems

A 1 kg ball at 4 m/s collides perfectly inelastically with a 2 kg ball at rest. Find the final velocity.
Two cars, 1000 kg and 1500 kg, collide and stick together. Velocities before collision: 8 m/s and 3 m/s. Find the final velocity.
A 0.5 kg puck at 2 m/s hits a 0.5 kg puck at rest. Find velocity after sticking together.
A 3 kg cart moving at 6 m/s collides inelastically with a 2 kg cart at 2 m/s. Find their final velocity if they stick together.
Two skaters, 40 kg and 60 kg, collide and stick. Velocities before collision: 1 m/s and -2 m/s. Find final velocity.