Impulse-Momentum Theorem
Reference for impulse-momentum theorem J = Ft = Δp = mΔv.
Covers elastic and inelastic collisions, conservation of momentum, and crash and rocket examples.
The Formula
J = F × Δt = Δp = m × Δv
The impulse-momentum theorem states that the impulse (force applied over time) equals the change in momentum. A larger force or a longer application time produces a greater change in velocity.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| J | Impulse | N·s (Newton-seconds) |
| F | Average force applied | N (Newtons) |
| Δt | Time the force is applied | s (seconds) |
| Δp | Change in momentum | kg·m/s |
| m | Mass of the object | kg |
| Δv | Change in velocity | m/s |
Example 1
A 0.15 kg baseball goes from 0 to 40 m/s after being hit. Find the impulse.
J = m × Δv = 0.15 × (40 - 0)
= 6 N·s
Example 2
A 1000 kg car stops from 20 m/s in 5 seconds. Find the average braking force.
Δp = m × Δv = 1000 × (0 - 20) = -20,000 kg·m/s
F = Δp / Δt = -20,000 / 5
= -4,000 N (the negative sign means the force opposes motion)
When to Use It
Use the impulse-momentum theorem when:
- Analyzing car crash forces and airbag design
- Calculating forces in sports (bat hitting ball, foot kicking ball)
- Designing shock absorbers and protective equipment
- Solving collision problems when time of contact is known
Key Notes
- Impulse-momentum theorem: J = Δp = mΔv: Impulse (J = FΔt) equals the change in momentum. A large force over a short time gives the same impulse as a small force over a long time.
- Units are equivalent: N·s = kg·m/s. Both are correct units for impulse and momentum — they are dimensionally identical.
- Conservation of momentum: In a closed system with no external forces, total momentum is conserved: p₁_before + p₂_before = p₁_after + p₂_after. This applies to all collisions.
- Elastic vs inelastic collisions: In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, only momentum is conserved — some KE converts to heat or deformation.
- Practical design insight: Car crumple zones and padded dashboards extend the collision time (Δt), which reduces the peak force even though the impulse (and thus Δp) stays the same.