Momentum & Impulse
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 7 practice questions.
Momentum is p = mv, a vector. A force changes it at the rate F = dp/dt. Over a time interval, the impulse J = ∫F dt equals the change in momentum, Δp, and it’s the area under the force–time graph.
1. Impulse
2. Direction and time
Momentum is a vector, so signs matter. A ball hitting a wall at 10 m/s and bouncing back at 8 m/s changes its velocity by 18 m/s, not 2.
A bounce takes more impulse than a stop, so it means a bigger force.
For the same Δp, a longer stopping time means a smaller force. That’s why airbags, padding and bending your knees on landing work.
For the same kinetic energy, the heavier object has more momentum: p = √(2mK).
3. Example, then try it
A 0.50 kg ball hits a wall at 6.0 m/s and bounces straight back at 4.0 m/s. Contact lasts 0.020 s. Find the impulse and the average force.
Taking the bounce direction as +: Δp = (0.50)(4.0) − (0.50)(−6.0) = 5.0 kg·m/s.
Favg = 5.0 ÷ 0.020 = 250 N, away from the wall.
A net force F(t) = 12t − 6t² (N) acts on a 3.0 kg object at rest from t = 0 to 2.0 s. Find the impulse, the final speed, the average force and the largest force.
Impulse: J = ∫02(12t − 6t²) dt = [6t² − 2t³] = 24 − 16 = 8.0 N·s.
Final speed: v = J/m = 8.0 ÷ 3.0 ≈ 2.7 m/s.
Average force: 8.0 ÷ 2.0 = 4.0 N.
Largest force: dF/dt = 12 − 12t = 0 at t = 1 s, where F = 12 − 6 = 6.0 N.
Answer: 8.0 N·s, 2.7 m/s, 4.0 N average, 6.0 N peak.
4. Common mistakes
An egg dropped onto a pillow survives, but one dropped from the same height onto the floor breaks. Why?
Both eggs go from the same speed to rest, so Δp is the same. The pillow stretches the stopping time, so Favg = Δp/Δt is smaller.
Ready to practice?
7 questions on Momentum & Impulse, each with a hint and an explanation for every wrong answer.