How to Calculate Momentum: Formula, Steps and Worked Examples

Momentum is one of those quantities that sounds complicated and turns out to be a single multiplication. Mass times velocity. That is the whole calculation. The part that trips people up is not the arithmetic, it is knowing which velocity to use, what to do when something changes direction, and how to handle two objects hitting each other.

This guide works through all of it: the formula, the units, the step by step method, and solved examples for straight line motion, collisions, change in momentum and impulse.

What Momentum Actually Measures

Momentum measures how hard it is to stop something that is already moving. A shopping trolley rolling at walking pace is easy to stop with one hand. A car rolling at the same speed is not, even though the speed is identical. The difference is mass.

Flip it around and the same idea holds. A tennis ball thrown gently is easy to catch. The same tennis ball fired from a machine at 200 km/h will sting, even though nothing about its mass has changed. The difference is velocity.

So momentum depends on both, and it depends on them equally. Double the mass and you double the momentum. Double the velocity and you double it again. That is why the formula is a straight multiplication rather than something with squares or fractions in it.

The Momentum Formula

p = mv

What Each Symbol Means

• p is momentum. The letter comes from the Latin petere, and physicists kept it because m was already taken by mass.
• m is mass, in kilograms. Not weight, and not grams.
• v is velocity, in metres per second. Velocity carries a direction with it, which matters more than it first appears.

The Units of Momentum

Momentum is measured in kilogram metres per second, written kg·m/s. There is no special name for this unit, no equivalent of the newton or the joule, so you write out the combination. You will also see it written as a newton second (N·s), which is the same thing because a newton is a kilogram metre per second squared. Both come from the SI base units, so either form is correct.

Two unit mistakes cause most wrong answers. Mass given in grams has to be divided by 1000 before you use it. Speed given in kilometres per hour has to be divided by 3.6 to get metres per second.

How to Calculate Momentum Step by Step

1. Write down the mass in kilograms. Convert from grams or tonnes first if you need to.
2. Write down the velocity in metres per second. Convert from km/h by dividing by 3.6.
3. Pick a positive direction and stick to it. Anything moving the other way gets a minus sign.
4. Multiply mass by velocity.
5. Write the unit and the direction. An answer of 24000 is incomplete. An answer of 24000 kg·m/s forward is finished.

Worked Examples

Example 1: A Moving Car

A car with a mass of 1200 kg travels at 20 m/s.

p = mv = 1200 × 20 = 24000 kg·m/s

Large numbers are normal here. Anything with the mass of a car carries momentum in the tens of thousands, which is exactly why a low speed collision still does serious damage.

Example 2: A Kicked Football

A football has a mass of 430 g and leaves the boot at 25 m/s. The mass has to go into kilograms first, so 430 g becomes 0.43 kg.

p = mv = 0.43 × 25 = 10.75 kg·m/s

Skip the conversion and you get 10750, which is a thousand times too big. This is the single most common error on momentum questions.

Example 3: A Bullet Against a Walking Person

A bullet of mass 10 g travels at 400 m/s. A person of mass 70 kg walks at 1.4 m/s.

• Bullet: p = 0.01 × 400 = 4 kg·m/s
• Person: p = 70 × 1.4 = 98 kg·m/s

The walking person has about 24 times more momentum than the bullet. That result surprises most people, and it is worth sitting with, because it shows momentum is not the same as damage. The bullet does far more harm because it delivers its momentum into a tiny area over a very short time. Momentum tells you about stopping. It does not tell you about destruction.

Momentum Has Direction, Not Just Size

Momentum is a vector. It has a size and a direction, and the direction is part of the answer rather than a footnote to it. A 1200 kg car at 20 m/s north and the same car at 20 m/s south have the same magnitude of momentum but opposite momentum vectors.

In one dimension you handle this with signs. Choose right as positive, and anything moving left gets a negative velocity and therefore negative momentum. The signs then do the work for you in every collision calculation that follows.

This is also why momentum can be negative while mass and speed never are. A negative momentum is not a smaller momentum. It is the same amount pointing the other way.

How to Calculate a Change in Momentum

Change in momentum is written with the Greek letter delta:

Δp = p_final - p_initial = mv_final - mv_initial

When the mass stays the same, which is almost always, you can factor it out and work with the velocities:

Δp = m(v_final - v_initial)

The order matters. Final minus initial, never the other way round, or your sign comes out backwards.

Worked Example: A Baseball Off the Bat

A baseball of mass 0.15 kg arrives at the bat at 40 m/s and leaves in the opposite direction at 45 m/s.

Take the outgoing direction as positive. The incoming velocity is then -40 m/s and the outgoing velocity is +45 m/s.

Δp = 0.15 × (45 - (-40)) = 0.15 × 85 = 12.75 kg·m/s

Notice the 85. Because the ball reversed direction, the change in momentum is bigger than either momentum on its own. Treat the incoming velocity as +40 and you get 0.15 × 5 = 0.75, which is wrong by a factor of 17. Reversal is where the signs earn their keep.

Impulse and the Impulse Momentum Theorem

Impulse is the force applied to something multiplied by how long it acts:

J = FΔt

And here is the result that connects the two ideas. Impulse equals change in momentum:

FΔt = Δp

This is the impulse momentum theorem, and it is just Newton’s second law rearranged. It is useful because it lets you find a force you could never measure directly, as long as you know the change in momentum and the contact time.

Worked Example: Force on the Baseball

The bat and ball were in contact for about 0.7 milliseconds, which is 0.0007 s. The change in momentum was 12.75 kg·m/s.

F = Δp / Δt = 12.75 / 0.0007 = about 18200 N

That is roughly the weight of two small cars, delivered in less than a thousandth of a second. The same equation read the other way explains airbags, crumple zones and crash mats. The change in momentum is fixed by the collision, so the only way to reduce the force is to stretch out the time.

Conservation of Momentum

In a closed system, with no outside forces acting, the total momentum before an event equals the total momentum after it:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Here u is velocity before and v is velocity after. This is the law of conservation of momentum, and it is the reason momentum is worth calculating at all. Energy can disappear into heat, sound and bent metal. Momentum has nowhere to go.

Worked Example: Two Carts That Stick Together

A 2 kg cart moving at 3 m/s hits a stationary 1 kg cart and the two lock together. How fast does the pair move afterwards?

Total momentum before: (2 × 3) + (1 × 0) = 6 kg·m/s

After the collision the combined mass is 3 kg, and momentum is unchanged at 6 kg·m/s.

v = p / m = 6 / 3 = 2 m/s

Check the energy if you want to see the difference between the two laws. Before: 9 J. After: 6 J. Three joules went into the sound of the crash and the deformation of the carts. The momentum did not change by a thousandth.

When Momentum Is Not Conserved

Momentum is only conserved when the system is closed. The moment an outside force gets involved, the total changes. A ball bouncing off a wall appears to break the law until you remember the wall is bolted to the Earth, and the Earth is part of the system. Friction and gravity do the same thing over longer timescales. In exam questions, “smooth”, “frictionless” and “isolated” are the words that tell you conservation applies.

Angular Momentum in One Minute

Spinning objects have their own version of momentum:

L = Iω

I is the moment of inertia, which depends on mass and how that mass is spread out from the axis. ω is angular velocity in radians per second. Angular momentum is conserved in the same way linear momentum is, which is why a figure skater speeds up when they pull their arms in. The mass has not changed, but moving it closer to the axis cuts the moment of inertia, so the angular velocity has to rise to keep L the same.

Four Mistakes That Cost Marks

1. Grams left as grams. Every mass goes into kilograms before it touches the formula.
2. km/h left as km/h. Divide by 3.6. A car at 72 km/h is a car at 20 m/s.
3. Ignoring direction in a reversal. If something bounces back, one of the two velocities is negative. Nearly every lost mark on change in momentum comes from here.
4. Confusing momentum with kinetic energy. Momentum is mv and is conserved in every collision. Kinetic energy is ½mv² and is only conserved in elastic ones.

If you want to go further, the rest of the physics guides work through the related quantities in the same step by step way.

Frequently Asked Questions

What is the momentum equation?

The momentum equation is p = mv, where p is momentum, m is mass in kilograms and v is velocity in metres per second. The answer comes out in kilogram metres per second (kg·m/s).

How do you calculate change in momentum?

Multiply the mass by the final velocity minus the initial velocity. Subtract the starting velocity from the final velocity, then multiply by the mass. If the object reversed direction, one of those velocities is negative.

What are the units of momentum?

Kilogram metres per second, written kg·m/s. The newton second (N·s) is an equivalent form, because one newton is one kilogram metre per second squared.

Is momentum a vector or a scalar?

Momentum is a vector. It has both a size and a direction, and the direction is part of the answer. In one dimension you show it with a plus or minus sign.

How is momentum different from force?

Momentum describes the motion an object already has. Force describes something acting on it to change that motion. They are linked by FΔt = Δp, so a force applied over time is what changes momentum.

How do you calculate impulse in physics?

Impulse is J = FΔt, the force multiplied by the time it acts for. It is also equal to the change in momentum, so if you know Δp you already know the impulse.

What is the law of conservation of momentum?

In a closed system with no outside forces, total momentum before an event equals total momentum after it. For two objects that is m1u1 + m2u2 = m1v1 + m2v2.

Can momentum be negative?

Yes. A negative momentum means the object is moving in the direction you chose as negative. The size of the momentum is the same as the positive version, it is simply pointing the other way.

How do you calculate momentum after a collision?

Add up the momentum of every object before the collision, then set that total equal to the momentum afterwards. If the objects stick together, divide the total momentum by the combined mass to get their shared velocity.

How do you calculate angular momentum?

Angular momentum is L = Iω, the moment of inertia multiplied by the angular velocity in radians per second. For a single point mass going round a circle it simplifies to L = mvr.