Linear Momentum & Its Conservation

Mass times velocity, added up over a system, refuses to change unless pushed from outside.

10 minUpdated 21 October 2026linear momentumconservation of momentumrecoil

Lesson at a glanceShow the key points

Linear momentum p = mv; for a system P = Σ mᵢvᵢ = M v(cm).
If the external forces on a system add to zero, its total momentum is conserved (in size and direction).
A system initially at rest that breaks into two parts sends them off with equal and opposite momenta: MV = −mv (recoil).
Momentum conservation follows from Newton's third law; it holds even when kinetic energy is not conserved.

Fire a rifle and it kicks your shoulder. Throw a ball off a skateboard and you roll backwards. Both are one law at work: the total momentum of an isolated system never changes.

Momentum

Definition · Linear momentum

For a particle,

\vec p = m\vec v

. For a system, add the particles’ momenta as vectors:

\vec P = \sum m_i\vec v_i

Because $\sum m_i\vec r_i = M\vec R_{cm}$ , differentiating gives a neat result: a system’s total momentum is its mass times the velocity of its centre of mass.

Momentum of a system(9.11)

\vec P = M\vec V_{cm}

Conservation of linear momentum

From the last lesson, if the external forces add up to zero then $\vec a_{cm} = 0$ , so $\vec V_{cm}$ and hence $\vec P$ are constant:

Note what is not required: kinetic energy may change wildly (explosions create it, crashes destroy it), and the internal forces may be of any kind (springs, friction, chemical, nuclear). As long as the outside world exerts no net force, momentum is fixed.

Interactive

Recoil: momentum before and after

Rifle and bullet start at rest: total momentum zero. After firing, the bullet's forward momentum is matched exactly by the rifle's backward momentum.

bullet momentum8.00kg·m/s

rifle momentum-8.00kg·m/s

rifle recoil speed2.00m/s

total0.00kg·m/s

Rifle mass

M

4.0kg

Bullet mass

m

0.020kg

Bullet speed

v

400m/s

Worked example

A rifle's kick

A 4.0 kg rifle fires a 20 g bullet at 400 m/s. With what speed does the rifle recoil (if held loosely)?

Worked example

A nucleus spits out an alpha particle

A uranium-238 nucleus at rest emits an alpha particle (mass 4 u) at 1.5 × 10⁷ m/s. Find the recoil speed of the remaining thorium-234 nucleus.

\text{initially at rest, splitting in two: } \quad M\vec V = -m\vec v

What to remember

1 $\vec p = m\vec v$ ; for a system $\vec P = \sum m_i\vec v_i = M\vec V_{cm}$ .
2Zero net external force ⇒ total momentum constant.
3A system at rest that splits sends the parts off with equal and opposite momenta.
4Momentum can be conserved even when kinetic energy is not.

Check yourself

An astronaut floating at rest throws a 2 kg tool at 3 m/s. She has mass 60 kg. Her recoil speed is…

During an explosion inside a shell, which is conserved?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchIntroduction to momentumKhan Academy ReadMomentumThe Physics Classroom ReadMomentum conservation principleThe Physics Classroom ReadIsolated systemsThe Physics Classroom ReadLinear momentumOpenStax University Physics §9.1 ReadConservation of linear momentumOpenStax University Physics §9.3 ReadMomentumHyperPhysics ReadConservation of momentumFeynman Lectures, Vol. I, Ch. 10

PreviousHow the Centre of Mass Moves Next Rocket Propulsion

Linear Momentum & Its Conservation

Mass times velocity, added up over a system, refuses to change unless pushed from outside.

10 minUpdated 21 October 2026linear momentumconservation of momentumrecoil

Lesson at a glanceShow the key points

Linear momentum p = mv; for a system P = Σ mᵢvᵢ = M v(cm).
If the external forces on a system add to zero, its total momentum is conserved (in size and direction).
A system initially at rest that breaks into two parts sends them off with equal and opposite momenta: MV = −mv (recoil).
Momentum conservation follows from Newton's third law; it holds even when kinetic energy is not conserved.

Momentum

Definition · Linear momentum

For a particle,

\vec p = m\vec v

. For a system, add the particles’ momenta as vectors:

\vec P = \sum m_i\vec v_i

Because

\sum m_i\vec r_i = M\vec R_{cm}

, differentiating gives a neat result: a system’s total momentum is its mass times the velocity of its centre of mass.

Momentum of a system(9.11)

\vec P = M\vec V_{cm}

Conservation of linear momentum

From the last lesson, if the external forces add up to zero then

\vec a_{cm} = 0

, so

\vec V_{cm}

and hence

\vec P

are constant:

Interactive

Recoil: momentum before and after

Rifle and bullet start at rest: total momentum zero. After firing, the bullet's forward momentum is matched exactly by the rifle's backward momentum.

bullet momentum8.00kg·m/s

rifle momentum-8.00kg·m/s

rifle recoil speed2.00m/s

total0.00kg·m/s

Rifle mass

M

4.0kg

Bullet mass

m

0.020kg

Bullet speed

v

400m/s

Worked example

A rifle's kick

A 4.0 kg rifle fires a 20 g bullet at 400 m/s. With what speed does the rifle recoil (if held loosely)?

Worked example

A nucleus spits out an alpha particle

A uranium-238 nucleus at rest emits an alpha particle (mass 4 u) at 1.5 × 10⁷ m/s. Find the recoil speed of the remaining thorium-234 nucleus.

\text{initially at rest, splitting in two: } \quad M\vec V = -m\vec v

What to remember

1 $\vec p = m\vec v$ ; for a system $\vec P = \sum m_i\vec v_i = M\vec V_{cm}$ .
2Zero net external force ⇒ total momentum constant.
3A system at rest that splits sends the parts off with equal and opposite momenta.
4Momentum can be conserved even when kinetic energy is not.

Check yourself

An astronaut floating at rest throws a 2 kg tool at 3 m/s. She has mass 60 kg. Her recoil speed is…

During an explosion inside a shell, which is conserved?

Momentum#

Conservation of linear momentum#

Check yourself

Keep exploring

Momentum#

Conservation of linear momentum#

Check yourself

Keep exploring

Momentum

Conservation of linear momentum

Momentum

Conservation of linear momentum