Momentum and impulse are two linked ideas in mechanics: momentum is the product of an object’s mass and velocity (p = mv), measured in kilogram-metres per second, while impulse is the change in momentum produced by a force acting over time (J = FΔt). Impulse equals the change in momentum an object experiences.
Snatch your hands back as you catch a cricket ball and it stings far less than catching it stiff-armed. Same ball, same speed, same change in motion — yet one catch hurts and the other barely registers. Why?
That gap is the whole story of momentum and impulse. Stretch a collision out over more time and the force drops; cram it into an instant and the force spikes. Airbags, crumple zones, a boxer rolling with a punch, and a rocket climbing to orbit all live inside this one idea.
What Is Momentum?
Picture a loaded lorry rolling at walking pace next to a tennis ball fresh off a serve. Which is harder to stop? The lorry — even crawling — because momentum depends on mass as much as on speed.
Momentum is the “quantity of motion” an object carries. Formally it is mass times velocity, written p = mv. Because velocity points somewhere, momentum is a vector: a ball rolling east and an identical ball rolling west carry equal-sized but opposite momenta.
Double the mass and you double the momentum. Double the velocity and you double it again. That is why a 10-tonne lorry creeping at 1 m/s and a 1 kg ball screaming along at 10,000 m/s carry the same momentum on paper — though you would happily catch neither.
The SI unit of momentum is the kilogram-metre per second (kg·m/s). One more thing matters: momentum tracks velocity, not speed, so direction counts and a change of sign signals a reversal. For the formal definition and unit, see Georgia State University’s HyperPhysics entry on momentum.
The Momentum and Impulse Formulas
Two compact equations carry most of this topic. The first defines momentum; the second defines impulse and ties it straight back to momentum.
- p — momentum, in kilogram-metres per second (kg·m/s)
- m — mass of the object, in kilograms (kg)
- v — velocity, in metres per second (m/s); a vector, so direction matters
- J — impulse, in newton-seconds (N·s), which is identical to kg·m/s
- F — average net force applied, in newtons (N)
- Δt — time interval over which the force acts, in seconds (s)
- Δp — change in momentum, equal to mvfinal − mvinitial, in kg·m/s
Notice that impulse and momentum share a unit. That is not a coincidence — it is the clue that impulse is a change in momentum, just measured through the force that caused it.
| Feature | Momentum (p) | Impulse (J) |
|---|---|---|
| What it describes | The motion an object already has | The change in that motion |
| Definition | Mass × velocity | Force × time interval |
| Formula | p = mv | J = FΔt = Δp |
| SI unit | kg·m/s | N·s (= kg·m/s) |
| Quantity type | Vector | Vector |
| Key link | The state of motion | Equals the change in momentum |
How Momentum and Impulse Work: The Impulse–Momentum Theorem
Where does J = Δp actually come from? Straight out of Newton’s second law, in four short steps.
- Start with the second law: F = ma.
- Acceleration is the rate of change of velocity: a = Δv/Δt.
- Substitute: F = m·Δv/Δt.
- Multiply both sides by Δt: F·Δt = m·Δv = Δp.
The left side, force multiplied by the time it acts, is the impulse. The right side is the change in momentum. So the impulse delivered to an object equals its change in momentum — the impulse–momentum theorem.
Read it the other way and it becomes a design tool: F = Δp/Δt. For a fixed change in momentum, the force is set entirely by the time you allow. Lengthen the time and the force shrinks.
That single rearrangement is why a longer collision is a gentler one — the reason airbags exist. The graph below makes it visible.
Two collisions, equal shaded areas: the same impulse and the same change in momentum, but the wider one needs far less peak force. Airbags simply widen the base.
Reading impulse off a force–time graph
Real forces are rarely constant — picture a boot meeting a ball, where the push swells and fades within milliseconds. For any such force, the impulse is still the area under the force–time graph, even when that shape is a curve.
For a constant force the area is just a rectangle, F × Δt. For the triangular spikes above it is ½ × base × height. The takeaway is the same either way: a wider, lower bump can hold the same area — the same impulse — as a tall, narrow one.
Want to feel the trade-off rather than just read it? The lab below lets you change the contact time and watch the peak force respond for a fixed change in momentum.
Momentum and Impulse in the Real World
Once you start looking, the impulse–momentum trade-off is everywhere — in safety engineering, in sport, and in how anything propels itself.
Airbags and crumple zones
A crash brings a body’s momentum to zero no matter what. An airbag and a crumple zone cannot change that drop in momentum — but they stretch it over a few tenths of a second instead of a few hundredths. Because F = Δp/Δt, multiplying the time by ten divides the peak force on the chest by ten.
Catching, landing, and rolling with a punch
A cricketer “gives” with the ball, drawing the hands back on the catch. A gymnast bends the knees on landing. A boxer rolls the head away from a blow. Each one buys extra time, and extra time means a softer force for the very same change in momentum.
Rockets and recoil
A rocket carries no road to push against, yet it accelerates in empty space. It throws exhaust gas backwards at high speed, and the forward momentum it gains exactly balances the backward momentum of that gas. The same bookkeeping explains a rifle’s kick. This is conservation of momentum, as NASA sets out for propulsion.
Sport: the follow-through
Coaches drill the follow-through for a reason rooted in physics. Keeping the club, bat, or racket on the ball for longer raises the impulse (F·Δt), and a bigger impulse means a bigger change in the ball’s momentum — so it leaves faster.
Newton’s cradle
That desk toy with the swinging steel balls is conservation of momentum on display. Lift one ball, release it into the row, and a single ball swings off the far end at nearly the same speed. The momentum travels cleanly along the line because the steel-on-steel collisions are almost perfectly elastic.
Conservation of Momentum
Here is one of the most powerful rules in physics: if no net external force acts on a system, its total momentum cannot change. Whatever the objects inside do to each other, the vector sum of their momenta before equals the vector sum after.
The reason sits in Newton’s third law. When two objects push on each other, the forces are equal and opposite, so they act for the same time and deliver equal-and-opposite impulses. Those impulses cancel, and the total momentum holds steady.
A 2 kg cart strikes and sticks to a 1 kg cart. The shared speed drops, but the total momentum is 6 kg·m/s before and after.
Elastic versus inelastic collisions
Momentum is conserved in every collision of an isolated system. Kinetic energy is choosier — it only survives intact in a perfectly elastic collision.
| Collision type | Momentum conserved? | Kinetic energy conserved? | Example |
|---|---|---|---|
| Elastic | Yes | Yes | Snooker balls; gas molecules |
| Inelastic | Yes | No — some lost to heat, sound, deformation | Most real car crashes |
| Perfectly inelastic | Yes | No — maximum possible lost | A bullet embedding in a block; train carriages coupling |
Common Misconceptions About Momentum and Impulse
A few sticky errors trip up almost everyone first time round. Clearing them is the fastest way to stop losing marks.
“Momentum and kinetic energy are basically the same.”
They are not. Momentum p = mv is a vector and grows in step with speed; kinetic energy ½mv² is a scalar and grows with the square of speed. Double an object’s speed and its momentum doubles, but its kinetic energy quadruples.
“Impulse is just another word for force.”
Impulse is force multiplied by the time it acts, with units of newton-seconds. A small force over a long time can deliver more impulse — more change in momentum — than a big force over a brief one.
“In a crash, the heavier vehicle hits the lighter one harder.”
By Newton’s third law the two forces are equal and opposite, so each vehicle feels the same size of force and the same impulse. The small car simply suffers a bigger change in velocity, because acceleration is force divided by mass.
“When something stops, its momentum is destroyed.”
Momentum is never destroyed, only transferred. A ball thudding into the ground hands its momentum to the Earth; you never notice because the Earth’s mass is colossal. Across an isolated system, the total always balances.
How Momentum and Impulse Relate to Other Physics
Momentum is a hub that connects much of mechanics, which makes it a great anchor for revision.
- Newton’s laws. The second law in its truest form is F = Δp/Δt, and the third law is what guarantees momentum is conserved. Start with Newton’s laws of motion if collisions feel shaky.
- Energy. Collisions are where momentum and energy part ways: momentum always balances, kinetic energy only does so when the collision is elastic.
- Velocity and direction. Because momentum is a vector, getting the signs right means being comfortable with velocity rather than speed.
