Section 1.5: Momentum and Impulse

This section introduces the concepts of linear momentum, impulse, and their relationship. Conservation of momentum in collisions is also discussed.

Key Equations & Concepts:
  • Linear Momentum: \( \vec{p} = m \vec{v} \)
  • Impulse: \( \vec{J} = \vec{F} \Delta t \)
  • Impulse-Momentum Theorem: \( \vec{J} = \Delta \vec{p} \)
  • Conservation of Momentum: \( \sum \vec{p}_{initial} = \sum \vec{p}_{final} \) (in isolated systems)
  • Elastic and Inelastic Collisions

Example 1

A 0.5 kg ball moving at 4 m/s collides with a wall and rebounds at 3 m/s. Find the impulse delivered to the ball.

Initial momentum: \( p_i = 0.5*4 = 2 \text{ kg·m/s} \)

Final momentum: \( p_f = 0.5*(-3) = -1.5 \text{ kg·m/s} \)

Impulse: \( J = \Delta p = p_f - p_i = -1.5 - 2 = -3.5 \text{ N·s} \)

Example 2

A 1 kg cart initially at rest is pushed with a force of 5 N for 4 s. Find the change in momentum.

Impulse: \( J = F \Delta t = 5*4 = 20 \text{ N·s} \)

Change in momentum: \( \Delta p = J = 20 \text{ kg·m/s} \)

Practice Problems

  1. A 2 kg object moving at 6 m/s is stopped by a force in 3 s. Find the force.
  2. Two carts collide elastically: m1 = 1 kg, v1 = 4 m/s; m2 = 2 kg, v2 = -1 m/s. Find final velocities.
  3. A 0.2 kg ball moving at 5 m/s hits a bat and rebounds at 3 m/s. Compute impulse.
  4. A 1.5 kg cart at rest is acted upon by 6 N for 2 s. Find final velocity.
  5. Two bodies collide inelastically: m1 = 3 kg, v1 = 2 m/s; m2 = 2 kg, v2 = -1 m/s. Find combined velocity.
  6. A 0.8 kg puck hits a wall and rebounds in 0.1 s. Initial speed 10 m/s, final -8 m/s. Find average force.
  7. 1 kg object moving at 3 m/s collides with stationary 2 kg object. Perfectly inelastic. Find final velocity.
  8. A 2 kg ball is accelerated from 0 to 10 m/s in 5 s. Find impulse.
  9. A cart of mass 3 kg moving at 2 m/s is slowed to rest in 4 s. Find force applied.
  10. Two objects collide elastically, masses 1 kg and 2 kg, velocities 5 m/s and -2 m/s. Find post-collision velocities.
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