Momentum is a moving object’s “quantity of motion” — how hard it is to stop. A charging rugby player and a slow-rolling truck can carry the same momentum. What makes it one of the most powerful ideas in physics is that in any collision or explosion, the total momentum stays exactly the same. Nothing is created, nothing lost — it’s simply passed around.
📘 What you need to know
Momentum is the product of mass and velocity: p = mv (units kg m s−1)
Momentum is a vector — it has direction, so it can be positive or negative
Conservation of momentum: total momentum before a collision equals total momentum after, if no external resultant force acts
In symbols: total p before = total p after
It applies to collisions and explosions alike, and momentum is always conserved
Opposing momenta can cancel, giving a net momentum of zero
Linear momentum
Any object with mass that is moving has momentum. Linear momentum is the momentum of an object moving in a straight line, and it’s defined simply as mass times velocity:
Momentump = mv
Here p is momentum in kg m s−1, m is mass in kg, and v is velocity in m s−1. Because it depends on velocity, momentum grows with both how heavy something is and how fast it’s going — which is why a light bullet and a heavy truck can both be dangerous.
Momentum has direction
Momentum is a vector: it points in the same direction as the velocity. That means you have to track its sign. Pick a positive direction — usually the direction of the initial motion — and anything moving the other way gets a negative momentum.
This matters most when something rebounds. A ball flying at a wall has positive momentum; bounce it straight back and its velocity reverses, so its momentum becomes negative. Same speed, opposite sign.
Moving toward the wall, the ball’s momentum is positive. After bouncing straight back at the same speed, its velocity — and so its momentum — is negative. The sign flips even though the speed doesn’t change.
WE 1
A tennis ball of mass 58 g travels to the right at 9.0 m s−1. Calculate its momentum. If it then rebounds off a wall at the same speed, what is its new momentum?
Step 1 — convert and apply p = mv (right = positive)
m = 58 g = 0.058 kg
p = 0.058 × 9.0 = 0.52 kg m s⁻¹Step 2 — after rebound the velocity reversesp = 0.058 × (−9.0) = −0.52 kg m s⁻¹before: +0.52 | after: −0.52 kg m s⁻¹The minus sign shows the ball is now travelling the opposite way — a big change in momentum from just a bounce.
Conservation of momentum
Here’s the principle that makes momentum so useful:
Conservation of linear momentum
The total momentum before a collision equals the total momentum after, provided no external resultant force acts
In short:
In symbols
total momentum before = total momentum after
Since momentum is a vector, you add the individual momenta with their signs. Opposing momenta can cancel, so a system can even have a total momentum of zero. And momentum, like energy, is always conserved — it never disappears, it just gets transferred between the objects involved.
The phrase to hold onto is “closed system.” Momentum is conserved as long as no external resultant force acts on the objects — the forces they exert on each other during the collision are internal and always cancel (that’s Newton’s third law). In an exam, if the question says a surface is frictionless or ignores air resistance, that’s your cue: total momentum before = total momentum after.
Collisions in action
Take the classic case: object A moving toward a stationary object B. They collide and stick together, moving off as one combined lump. Momentum conservation lets you find their shared final speed without knowing anything about the messy forces during impact — you only need the before and after.
A moves toward stationary B, they collide and stick, and move off together at a shared velocity v. The total momentum before the collision equals the total momentum after.
🛠️ Solving a conservation problem
Choose a positive direction and note the mass and velocity of each object.
Write the total momentum before: add each mv, with signs. A stationary object contributes zero.
Write the total momentum after: if objects stick, treat them as one combined mass.
Set before = after and solve for the unknown velocity.
WE 2
A trolley A of mass 2.0 kg moves at 3.0 m s−1 and collides with a stationary trolley B of mass 4.0 kg. The trolleys stick together on impact. Calculate their common velocity afterwards.
Step 1 — total momentum before (right = positive)pbefore = (2.0 × 3.0) + (4.0 × 0) = 6.0 kg m s⁻¹Step 2 — total momentum after (stuck, combined mass 6.0 kg)
pafter = (2.0 + 4.0) × v = 6.0v
Step 3 — set before = after6.0 = 6.0v → v = 1.0 m s⁻¹v = 1.0 m s⁻¹ (to the right)The combined trolleys move slower than A alone did — the same momentum is now shared across three times the mass.
WE 3
A car of mass 1200 kg travelling at 8.0 m s−1 collides with a stationary van of mass 3000 kg. Just after the collision the car continues forward at 2.0 m s−1. Calculate the velocity of the van.
Step 1 — total momentum beforepbefore = 1200 × 8.0 = 9600 kg m s⁻¹
(van is at rest, so contributes 0)
Step 2 — total momentum after = same
9600 = (1200 × 2.0) + (3000 × v)
9600 = 2400 + 3000vStep 3 — solve for vv = 7200 ÷ 3000v = 2.4 m s⁻¹The momentum the car lost (9600 → 2400) is exactly the momentum the van gained — conservation in action.
💡 Top tips
Pick a positive direction first and keep the signs consistent all the way through.
A stationary object has zero momentum — but don’t forget its mass appears again after the collision.
When objects stick together, treat them as a single combined mass afterwards.
Convert grams to kilograms before using p = mv.
Momentum is always conserved in a closed system, even when kinetic energy isn’t.
Quick recap: momentum is p = mv, a vector, so track its sign. In a closed system with no external resultant force, total momentum before a collision equals total momentum after. Add momenta with signs, treat stuck objects as one combined mass, and solve before = after for the unknown.
⚠ Common mistakes
Ignoring the sign of momentum — a rebounding object’s momentum goes negative
Forgetting that a stationary object still has mass that matters after the collision
Leaving mass in grams instead of converting to kilograms
Assuming kinetic energy is conserved too — only momentum is always conserved
Applying conservation when an external force (like friction) acts — it only holds for a closed system
Momentum is conserved — but what about the energy in a collision? Sometimes kinetic energy survives intact, sometimes it’s lost to heat and sound. That distinction splits collisions into two types, and it’s exactly where we go next: Collisions & Explosions in One Dimension, and the difference between elastic and inelastic.
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