Section 5.3: Conservation of Momentum in Collisions

The conservation of momentum states that in an isolated system, the total momentum before a collision equals the total momentum after the collision:

\( \sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}} \)

This principle applies to both elastic and inelastic collisions, though kinetic energy is only conserved in elastic collisions.

Types of Collisions:
  • Elastic: Momentum and kinetic energy conserved.
  • Inelastic: Momentum conserved, kinetic energy not conserved.
  • Perfectly inelastic: Colliding objects stick together.

Example 1

A 2 kg cart moving at 3 m/s collides head-on with a 3 kg cart at rest. They stick together. Find final velocity.

Total initial momentum: \( p_i = 2*3 + 3*0 = 6 \text{ kg·m/s} \)

Mass after collision: \( m_{\text{total}} = 2 + 3 = 5 \text{ kg} \)

Final velocity: \( v_f = p_i / m_{\text{total}} = 6 / 5 = 1.2 \text{ m/s} \)

Example 2

Two ice skaters, 50 kg and 70 kg, push off each other. If the 50 kg skater moves at 2 m/s, find the velocity of the 70 kg skater.

Initial momentum: \( 0 \) (they start at rest)

Let \( v_2 \) be velocity of 70 kg skater: \( 50*2 + 70*v_2 = 0 \)

\( v_2 = -100/70 \approx -1.43 \text{ m/s} \)

Practice Problems

  1. Two carts, 3 kg and 5 kg, collide elastically. Find final velocities if initial velocities are 4 m/s and 0 m/s.
  2. A 2 kg ball moving 5 m/s strikes a 3 kg ball at rest. Perfectly inelastic collision. Find final velocity.
  3. A 1 kg puck hits a 2 kg puck at rest elastically. Find velocities after collision.
  4. Two skaters, 60 kg and 40 kg, push off each other. Find velocities if momentum conserved.
  5. A 0.2 kg ball moving 10 m/s collides with 0.3 kg ball at rest. Perfectly inelastic. Find final velocity.
  6. Two carts of equal mass 4 kg collide elastically: one moves at 6 m/s, other at -2 m/s. Find final velocities.
  7. A 3 kg ball moving 8 m/s collides inelastically with 2 kg ball at rest. Find final velocity.
  8. Two skaters with masses 45 kg and 55 kg start from rest. They push off each other and move in opposite directions. Find velocities if 45 kg skater moves at 1.5 m/s.
  9. Ball A, 0.5 kg at 4 m/s collides elastically with 0.5 kg ball B at rest. Find velocities after collision.
  10. A car of 1000 kg moving 20 m/s collides inelastically with a 1500 kg car at rest. Find final velocity.
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