The Memory Trick
💡 J = FΔt = Δp
Impulse is the product of force and the time over which it acts, and it always equals the resulting change in momentum: J = FΔt = Δp = mΔv. This means the same change in momentum can be achieved by a large force over a short time, or a smaller force spread over a longer time.
Why It Works
This is exactly why airbags, crumple zones, and bending your knees when landing all reduce injury — none of them change the total momentum that has to be absorbed, but all of them extend the time (Δt) over which it happens, which reduces the peak force (F) felt by the body.
Step by Step
Applying the Impulse-Momentum Theorem
1
Impulse equals momentum change — always
Whatever the details of the collision or interaction, the impulse delivered exactly equals the object's change in momentum. This is really just Newton's Second Law rearranged over a time interval.
A tennis racket applying force to a ball for a fraction of a second changes the ball's momentum by exactly J = FΔt.
2
Extending time reduces peak force
For a FIXED momentum change (like stopping a moving car), stretching out the time it takes to stop reduces the average force required.
Crumple zones increase the time a car takes to come to a stop during a crash, lowering the force transmitted to passengers compared to an instant, rigid stop.
3
Units confirm the relationship
Impulse is measured in N·s, and momentum in kg·m/s — these are actually the same units, since N = kg·m/s², making N·s = kg·m/s.
A 5 N·s impulse applied to a 2 kg object produces Δv = 5/2 = 2.5 m/s of velocity change.
🏥 Worked Example
A 0.15 kg baseball traveling at 40 m/s is caught by a glove and brought to rest in 0.05 seconds. What average force does the glove exert on the ball?
1
Calculate the momentum change: Δp = mΔv = 0.15 × (0 - 40) = -6 kg·m/s.
2
Apply J = FΔt = Δp: F × 0.05 = -6, so F = -6/0.05 = -120 N.
3
Interpret: the glove exerts 120 N on the ball (negative sign just indicates direction, opposite the ball's original motion) — if the catch instead took 0.5 seconds, the force would only need to be 12 N.
📌 Exam Application
Exams test correctly calculating Δp = mΔv, applying J = FΔt = Δp to solve for any missing variable, and explaining real-world safety applications in terms of extending Δt to reduce force.
⚠️ Most Common Impulse and Momentum Mistakes
The most common trap is forgetting that Δv (and therefore Δp) requires a direction — stopping an object moving at 40 m/s means Δv = 0 - 40 = -40 m/s, not +40 m/s, which changes the sign (but not magnitude) of the resulting force.
✓ Quick Self-Test
1) Write the impulse-momentum theorem. J = FΔt = Δp = mΔv. 2) Why do airbags reduce injury in a car crash, given that they don't change how much the passenger's momentum changes? They increase the time (Δt) over which that momentum change happens, which reduces the peak force required. 3) What are the units of impulse, and how do they relate to the units of momentum? N·s, which are equivalent to kg·m/s — the same units as momentum. 4) A 1500 kg car traveling at 20 m/s is stopped by a wall in 0.1 seconds. What average force does the wall exert? F = Δp/Δt = (1500×20)/0.1 = 300,000 N. 5) If the same car instead stopped over 2 seconds (like braking normally), what force would be required? F = (1500×20)/2 = 15,000 N — dramatically less, since the time is much longer.
Next Lesson
Fluid Mechanics Essentials
→
← All Mechanics Lessons