Momentum Formula: p = mv, Unit, Examples and Calculator

The momentum formula is $p = mv$ : linear momentum equals mass times velocity. A 1000 kg car moving at 20 m/s carries 20 000 kg m/s of momentum, while a 70 kg runner at the same speed carries only 1400 kg m/s, which is why the car is so much harder to stop.

Momentum is sometimes called the quantity of motion. It joins two ideas you already know, how much matter is moving and how fast it is moving, into one vector. It sits at the centre of the laws of motion because Newton's second law, in its full form, says force is the rate at which momentum changes.

Quick facts

Formula
$\vec{p} = m\vec{v}$
SI unit
kg m/s, which equals N s
Dimensional formula
[M¹ L¹ T⁻¹]
Type
Vector, in the direction of the velocity
Link with kinetic energy
$p = \sqrt{2mK}$
Link with force
$F = \frac{\Delta p}{\Delta t}$

Linear momentum formula: what p, m and v mean

Symbols in p = mv
Symbol	Meaning	SI unit
$p$	Linear momentum of the body	kg m/s
$m$	Mass, measured in kilograms	kg
$v$	Velocity, speed with a direction	m/s

Because velocity is a vector and mass is a positive scalar, momentum always points the same way as the velocity. If you choose east as positive, a trolley moving west at 3 m/s with mass 2 kg has $p = 2 \times (-3) = -6\ \text{kg m/s}$ . Keep that sign. It matters as soon as two bodies interact.

The word linear separates this from angular momentum, $L = I\omega$ , which describes spinning. When a question just says momentum, it means linear momentum.

Worked examples using the formula for momentum in physics

Example 1: a rolling ball

A 6.0 kg ball rolls at 4.0 m/s. Find its momentum.

$p = mv = 6.0 \times 4.0 = 24\ \text{kg m/s}$

Example 2: truck and bullet, with km/h

Compare the momentum of a 5000 kg truck at 72 km/h with a 10 g bullet at 500 m/s.

Truck: at 20 m/s (that is 72 km/h), p = 5000 × 20 = 1.0 × 10⁵ kg m/s.

Bullet: 10 g = 0.010 kg, so p = 0.010 × 500 = 5.0 kg m/s.

The slow truck has 20 000 times the momentum of the fast bullet. High speed alone does not mean large momentum.

Example 3: total momentum of two carts

Cart A (2.0 kg) moves right at 3.0 m/s. Cart B (1.0 kg) moves left at 4.0 m/s. Find the total momentum.

Take right as positive: p_A = 2.0 × 3.0 = 6.0 kg m/s and p_B = 1.0 × (−4.0) = −4.0 kg m/s.

$p_{\text{total}} = 6.0 + (-4.0) = 2.0\ \text{kg m/s, to the right}$

Adding the sizes, 10 kg m/s, would be wrong because the carts move in opposite directions.

Total momentum formula for a system

For several bodies, the total momentum is the vector sum $\vec{P} = m_1\vec{v}_1 + m_2\vec{v}_2 + \dots$ . In one dimension, add with signs as in Example 3. In two dimensions, add the x parts and y parts separately, then combine them with Pythagoras.

The total can also be written as $\vec{P} = M\vec{v}_{cm}$ , where M is the total mass and the velocity is that of the center of mass (the centre of mass in Indian books). When no outside force acts, this total stays fixed. That idea is developed on the conservation of momentum formula page and used for guns on the recoil velocity page.

Momentum, kinetic energy and force

Kinetic energy and momentum both depend on mass and speed, so one gives the other: $K = \frac{p^2}{2m}$ , or $p = \sqrt{2mK}$ . For two bodies with the same kinetic energy, the heavier one has more momentum. For two with the same momentum, the lighter one has more kinetic energy.

Example 4: momentum from kinetic energy

A 2.0 kg block has 100 J of kinetic energy. Find its momentum.

$p = \sqrt{2mK} = \sqrt{2 \times 2.0 \times 100} = \sqrt{400} = 20\ \text{kg m/s}$

Force enters through the second law: $F = \frac{\Delta p}{\Delta t}$ . A force acting for a time changes the momentum by $F\Delta t$ , the impulse. Worked cases of that step are collected under the change in momentum formula.

Formula for linear momentum: unit and dimensions

Multiplying kg by m/s gives kg m/s. Since a newton is kg m/s², a newton second is kg m/s² × s = kg m/s, so the two units are the same thing. Use kg m/s for momentum and N s for impulse by habit, but either is correct.

In dimensions, [M] × [L T⁻¹] gives [M¹ L¹ T⁻¹]. See the dimensional formula of momentum page for the derivation and a list of quantities with the same dimensions.

Common mistakes with momentum

Treating momentum as a scalar. Two equal and opposite momenta add to zero, not to twice the value.
Using speed for a change of direction. A ball hitting a wall at 5 m/s and bouncing back at 5 m/s changes momentum by 2mv, not zero.
Leaving mass in grams. Convert to kilograms before you multiply.
Mixing up p = mv and K = ½mv². Momentum grows in step with speed; kinetic energy grows with its square.

Write the sign of every velocity before you start. Most marks lost on momentum questions are sign errors, not arithmetic.

Frequently asked questions

What is the formula of momentum?

Momentum is mass times velocity, p = mv. With m in kg and v in m/s, the product comes out in kg m/s automatically.

Which SI unit does momentum have?

Kilogram metre per second, written kg m/s. The newton second, N s, is exactly the same unit.

Is momentum a vector quantity?

Yes. Momentum has the same direction as the velocity, so it can be positive or negative along a chosen axis.

Can a body have momentum without kinetic energy?

No. If p is not zero then v is not zero, so K = ½mv² is not zero. A body can have kinetic energy while a system of bodies has zero total momentum, though.

What is the difference between momentum and inertia?

Inertia is the tendency to resist a change in motion and is measured by mass alone. Momentum depends on both mass and velocity, so a body at rest has inertia but no momentum.