The Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse of the net force on an object equals the change in its momentum: F_netΔt = mv_f − mv_i. In brief: for a given change in momentum, a longer contact time means a smaller average force. This is why airbags, crumple zones, helmets, padded floors and bending the knees on landing reduce the force of an impact. Conversely, a short, sharp contact gives a large force, as when a karate expert breaks a board.

The theorem follows from Newton’s second law and is the key to analyzing impacts and collisions where forces are large and vary during the short contact. This guide states and derives the theorem, explains how it is used in safety design and sports, compares the cases in a table, works through numerical examples and corrects common mistakes.

What does the theorem say?

Newton’s second law in terms of momentum is F_net = Δp ÷ Δt. Multiplying both sides by Δt gives F_netΔt = Δp. The left side is the impulse J of the net force, and the right side is the change in momentum. If the force varies, F_net is the average force (or the impulse is the area under the force-time graph). In one dimension, the signs must be used for directions. The theorem holds regardless of the details of the collision, so it can be used when the force during the contact is complicated or unknown.

Why does contact time matter?

For a given change in momentum Δp, the product FΔt is fixed. If the stopping time is increased, the average force falls in proportion. Applications:

Airbags and seat belts extend the time over which a passenger’s momentum is reduced, from a few milliseconds to a longer time.
Crumple zones in cars deform and lengthen the collision time.
Bending the knees when landing, rolling on the ground (parachute landing), and padded mats in gymnastics increase the stopping time.
Catching a ball by moving the hands backward with the ball.
Helmets with foam liners.
Following through in golf or tennis increases the time of contact and so increases the impulse and the final speed of the ball. A sharp, short force is used when the aim is a large force, such as a hammer on a nail.

Average force on the 70 kg driver in the worked example. Both rectangles have the same area, the impulse 1,050 N·s, because the change in momentum is the same. Stopping in 0.050 s needs 21,000 N, but stretching the stop to 0.15 s with an airbag cuts the average force to 7,000 N.

Short and long contact times compared

Same change in momentum, different stopping times
Stopping time	Average force	Example
Short	Large	Landing on concrete with straight legs
Long	Small	Landing on a mattress, airbag

Worked example: airbags and a falling egg

Without an airbag. A driver of mass 70 kg moving at 15 m/s is stopped by the steering wheel in 0.050 s. Δp = 0 − 70 × 15 = −1,050 kg·m/s. F_avg = −1,050 ÷ 0.050 = −21,000 N, about 31 times the driver’s weight (686 N).
With an airbag. The airbag extends the stopping time to 0.15 s. F_avg = 1,050 ÷ 0.15 = 7,000 N, three times smaller, although still large (this is why seat belts are also needed).
Egg drop. An egg of mass 0.060 kg is dropped from 2.0 m. Speed on impact: v = √(2gh) = √(2 × 9.8 × 2.0) = 6.3 m/s, so p = 0.060 × 6.3 = 0.38 kg·m/s. On a hard floor with a stopping time of 0.002 s, F = 0.38 ÷ 0.002 = 190 N (the egg breaks). On a thick foam pad with a stopping time of 0.05 s: F = 0.38 ÷ 0.05 = 7.6 N (it survives).
Jumping. A person of mass 60 kg lands at 4.0 m/s. With stiff legs, the stopping time is 0.020 s: F = 60 × 4.0 ÷ 0.020 = 12,000 N. With bent knees, taking 0.20 s: F = 1,200 N (including the person’s weight, the total force from the floor is 1,200 + 588 = 1,790 N).

Where does the theorem come from?

Newton’s original statement of the second law was in terms of the change of momentum, so the impulse-momentum theorem is essentially his law in integrated form. Crumple zones were patented by Mercedes-Benz engineer Béla Barényi in 1952, and the airbag was developed from patents of the 1950s, becoming common in cars in the 1980s and 1990s.

Which mistakes are common?

Mistake 1: Thinking that a longer time reduces the impulse

The impulse (change in momentum) is fixed. A longer time reduces the average force.

Mistake 2: Forgetting the sign of the change

With a rebound, the final momentum has the opposite sign and Δp is larger than the initial magnitude.

Mistake 3: Using the maximum force as the average

The theorem gives the average force. The peak force during the impact can be higher.

Mistake 4: Ignoring weight when the contact time is long

For long contact times, the net force includes weight (F_floor − mg).