Section 5.1: Linear Momentum and Impulse

Momentum is a measure of the motion of an object and is defined as:

\( \vec{p} = m \vec{v} \)

where \(m\) is mass and \(\vec{v}\) is velocity. Momentum is a vector quantity.

Impulse is the change in momentum due to a force applied over a time interval:

\( \vec{J} = \vec{F} \Delta t = \Delta \vec{p} \)

Example 1

A 5 kg ball moving at 4 m/s is struck by a bat and its velocity reverses to 3 m/s. Find the impulse delivered by the bat.

Initial momentum: \( p_i = 5 \times 4 = 20 \text{ kg·m/s} \)

Final momentum: \( p_f = 5 \times (-3) = -15 \text{ kg·m/s} \)

Impulse: \( J = p_f - p_i = -15 - 20 = -35 \text{ N·s} \)

Example 2

A 2 kg cart initially at rest is pushed with a constant 6 N force for 3 s. Find the final momentum and velocity.

Impulse: \( J = F \Delta t = 6 \times 3 = 18 \text{ N·s} \)

Final momentum: \( p_f = 18 \text{ kg·m/s} \)

Final velocity: \( v_f = \frac{p_f}{m} = \frac{18}{2} = 9 \text{ m/s} \)

Practice Problems

  1. A 0.5 kg ball moving at 6 m/s is stopped by a hand in 0.1 s. Find the average force.
  2. A 3 kg cart moving at 2 m/s is hit to reach 5 m/s. Find the impulse applied.
  3. A 4 kg object at rest is pushed by 10 N for 2 s. Determine final velocity.
  4. A 6 kg object moving at 3 m/s collides head-on with a 4 kg object at 2 m/s. Compute total momentum before collision.
  5. A baseball of mass 0.145 kg moving at 40 m/s is hit back at 50 m/s. Find impulse.
  6. Car of 1200 kg accelerates from 0 to 20 m/s in 10 s. Find average force.
  7. A puck of mass 0.2 kg is at rest, then hit by a 10 N force for 0.5 s. Find final velocity.
  8. Impulse required to stop a 7 kg object moving at 3 m/s.
  9. Truck of mass 1500 kg moving at 25 m/s collides and stops. Find impulse.
  10. A 0.8 kg ball moving at 12 m/s is caught and brought to rest in 0.2 s. Compute average force.
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