The Momentum Formula, Explained with Examples
By ToolNimba Editorial Team June 25, 2026 7 min read
Quick answer
Momentum = mass times velocity, written as p = m x v. Here m is the mass in kilograms and v is the velocity in metres per second, so momentum is measured in kilogram metres per second (kg m/s). Momentum is a vector, which means it points in the same direction as the velocity. Heavier or faster objects carry more momentum.
Momentum is a measure of how much motion an object has and how hard it is to stop. A loaded truck rolling slowly and a small car speeding can both be tough to bring to rest, and momentum tells you exactly how much. In this guide you will see the formula, the units, how impulse changes momentum, why total momentum stays the same in collisions, several worked examples, and the most common mistakes to avoid.
What is momentum?
Momentum is the quantity of motion in a moving object. Anything with mass that is moving has momentum, and a stationary object has none. The amount depends on two things: how heavy the object is and how fast it is going. A bowling ball moving slowly can have the same momentum as a tennis ball moving very fast, because what matters is mass and velocity multiplied together.
Momentum is a vector quantity, which means it has both size and direction. The direction of the momentum is always the same as the direction of the velocity. A car moving north and an identical car moving south at the same speed have equal amounts of momentum, but in opposite directions, so they are not the same momentum. If you want to brush up on the difference between speed and velocity, see the velocity formula guide.
The momentum formula
The standard equation for the momentum of a moving object is:
Momentum
p = m x v, where p is the momentum in kilogram metres per second, m is the mass in kilograms, and v is the velocity in metres per second.
Each symbol has a specific meaning and a specific unit, and getting the units right is half the battle. The table below lays them out.
What each symbol means
| Symbol | Quantity | SI unit |
|---|---|---|
| p | Momentum | kilogram metre per second (kg m/s) |
| m | Mass | kilogram (kg) |
| v | Velocity | metre per second (m/s) |
Momentum has no special named unit of its own. It is simply kilograms multiplied by metres per second, written kg m/s. So if you plug a mass in kilograms and a velocity in metres per second into the formula, the answer lands in kg m/s automatically with no extra conversion needed. The letter p is used for momentum because m was already taken by mass.
How to calculate momentum step by step
Whatever the numbers, the method is always the same three steps.
- Make sure the mass is in kilograms and the velocity is in metres per second. Convert first if needed.
- Multiply the mass by the velocity.
- Attach the direction. The answer is in kg m/s and points the same way as the velocity.
Mixing units is the number one source of wrong answers. If a velocity is given in kilometres per hour, divide by 3.6 to get metres per second first. If a mass is given in grams, divide by 1000 to get kilograms. A quick velocity calculator can help when you only know distance and time and need the velocity before you can find momentum.
Worked example: a moving car
A car with a mass of 1000 kg is travelling east at 20 m/s. Find its momentum.
- Check the units: mass is 1000 kg and velocity is 20 m/s, both in SI units already.
- Multiply mass by velocity: 1000 x 20 = 20000.
- Attach the direction: the momentum is 20000 kg m/s to the east.
Now compare a heavy truck. A 5000 kg truck moving at just 8 m/s has a momentum of 5000 x 8 = 40000 kg m/s, double that of the faster car. This shows how a heavy object moving slowly can carry more momentum than a lighter object moving quickly.
Worked example: a thrown ball
A baseball with a mass of 0.145 kg is thrown at 40 m/s. Find its momentum.
- List the values: m = 0.145 kg, v = 40 m/s.
- Multiply mass by velocity: 0.145 x 40 = 5.8.
- State the answer: the momentum is 5.8 kg m/s in the direction the ball was thrown.
Impulse and the impulse-momentum theorem
Momentum does not change on its own. To change an object's momentum you have to apply a force over a period of time, and that combination is called impulse. The impulse-momentum theorem ties them together neatly:
Impulse-momentum theorem
Impulse = change in momentum = force x time, written J = F x t = delta p. A bigger force, or the same force applied for longer, produces a bigger change in momentum.
This explains a lot of everyday physics. Air bags, crumple zones, and crash mats all work by stretching out the time over which momentum changes. The change in momentum is fixed by how fast you were moving, so spreading it over a longer time means a smaller force, which means less damage. The same idea is why you bend your knees when landing a jump: a longer stopping time means a gentler force. Force, by the way, comes from acceleration, which you can review in the acceleration formula guide.
Momentum, impulse, and the related quantities
| Quantity | Formula | SI unit |
|---|---|---|
| Momentum | p = m x v | kg m/s |
| Impulse | J = F x t | newton second (N s) |
| Change in momentum | delta p = J = F x t | kg m/s |
One neat detail: a newton second and a kilogram metre per second are the same unit in disguise, which is exactly why impulse equals a change in momentum. The two sides of the theorem balance perfectly.
Conservation of momentum
The most powerful idea in this whole topic is conservation. In a closed system with no external force acting, the total momentum stays the same: total momentum before equals total momentum after. This holds true no matter what happens inside the system, which is why physicists use it to analyse collisions and explosions.
For two objects colliding, conservation of momentum is written like this:
Conservation of momentum
m1 x v1 + m2 x v2 (before) = m1 x v1prime + m2 x v2prime (after). The total of mass times velocity for both objects is the same before and after the collision.
Because momentum is a vector, direction matters here. You usually pick one direction as positive and treat the opposite direction as negative. An object moving left might be entered as a negative velocity so the totals add up correctly. This is the single most common reason a conservation problem goes wrong, so keep your signs consistent from start to finish.
Worked example: two carts collide and stick
A 2 kg cart moving at 3 m/s strikes a stationary 1 kg cart, and the two stick together. How fast do they move afterwards?
- Find the total momentum before: 2 x 3 plus 1 x 0 = 6 kg m/s.
- After the collision they move together, so their combined mass is 2 plus 1 = 3 kg.
- Set total momentum after equal to before: 3 x v = 6.
- Solve for the final velocity: v = 6 divided by 3 = 2 m/s.
- State the answer: the joined carts move off at 2 m/s in the original direction.
Notice the total momentum was 6 kg m/s before and 6 kg m/s after. Momentum was conserved even though the carts changed speed and joined together. This is the engine behind everything from billiards to rocket propulsion.
Common mistakes to avoid
- Treating momentum as a scalar. Momentum has direction. Two objects moving opposite ways do not simply add; one velocity must be negative.
- Using the wrong units. Mass must be in kilograms and velocity in metres per second for the answer to come out in kg m/s. Convert grams or km/h first.
- Confusing momentum with kinetic energy. Momentum is p = m x v, while energy is 1/2 x m x v squared. See the kinetic energy formula for the difference.
- Forgetting the system must be closed. Conservation only holds when no external force acts. Friction, gravity, or a push from outside will change the total.
- Dropping the sign in collisions. Keep one direction positive and the other negative throughout, or your before and after totals will not match.
Good to know
Momentum and kinetic energy are related but not the same. Momentum depends on velocity to the first power and is always conserved in a closed system. Kinetic energy depends on velocity squared and is only conserved in special collisions called elastic collisions. When two cars crash and crumple, momentum is still conserved but a lot of kinetic energy is lost to heat, sound, and bending metal. Distinguishing the two quantities is one of the most useful skills in physics.
Conservation of momentum is also why rockets work in the vacuum of space. The rocket pushes exhaust gases backwards, and to keep the total momentum unchanged the rocket itself must move forwards. No air or ground is needed. Together with velocity and acceleration, momentum gives you a complete picture of how and why objects move and interact.
Frequently asked questions
What is the momentum formula?
The momentum formula is p = m x v, meaning momentum equals mass times velocity. Mass m is in kilograms and velocity v is in metres per second, so momentum is measured in kilogram metres per second (kg m/s). Momentum is a vector pointing the same way as the velocity.
What are the units of momentum?
The SI unit of momentum is the kilogram metre per second (kg m/s). It has no special name of its own. You get kg m/s automatically when mass is in kilograms and velocity is in metres per second, since momentum is just the two multiplied together.
Is momentum a vector or a scalar?
Momentum is a vector, so it has both size and direction. Its direction is always the same as the velocity. Two objects moving in opposite directions at the same speed have equal amounts of momentum but pointing opposite ways, so their momenta are not identical.
What is the impulse-momentum theorem?
The impulse-momentum theorem says impulse equals the change in momentum, written J = F x t = delta p. Applying a force for a length of time changes an object's momentum. A bigger force or a longer time produces a bigger change, which is how air bags and crumple zones reduce impact forces.
What is conservation of momentum?
Conservation of momentum means that in a closed system with no external force, the total momentum stays the same: total before equals total after. For a collision, m1 x v1 + m2 x v2 before equals m1 x v1prime + m2 x v2prime after. It is the key tool for analysing collisions.
How is momentum different from kinetic energy?
Momentum is p = m x v and depends on velocity to the first power, while kinetic energy is 1/2 x m x v squared and depends on velocity squared. Momentum is a vector and is always conserved in a closed system; kinetic energy is a scalar and is only conserved in elastic collisions.