Linear Momentum of Particle Systems

When physicists study objects in the real world, they rarely deal with a single point particle in isolation. A football spinning through the air, a cluster of billiard balls after a break shot, or even the atoms vibrating inside a solid rod are all examples of systems made up of many particles interacting with one another. To make sense of the motion of such systems without tracking every individual particle separately, physicists introduce the concept of the total linear momentum of a system — one of the most powerful and far-reaching ideas in all of classical mechanics. The total linear momentum of a multi-particle system is simply the vector sum of the individual momenta of every particle that belongs to the system. Because momentum is a vector, direction matters: particles moving in opposite directions can cancel each other's contributions. This total quantity obeys a beautifully simple rule — it changes only when a net external force acts on the system. Internal forces between particles, no matter how large, cannot change the system's total momentum. This principle, known as conservation of linear momentum, connects directly to Newton's third law and underlies everything from rocket propulsion to the physics of collisions. Understanding system momentum also provides the foundation for studying the center of mass — the single point that moves as if the entire mass of the system were concentrated there and all external forces acted upon it. Together, total momentum and center-of-mass motion allow us to analyze complex rotating and translating rigid bodies without losing track of individual particles. This concept sits at the heart of the Rotational Motion and Rigid Body Dynamics unit, linking translational and rotational descriptions of motion.

Key takeaways
  • Momentum of a Single Particle
  • Summing Over All Particles
  • Connecting to the Center of Mass
  • Newton's Second Law for Systems
  • Conservation of Momentum

Momentum of a Single Particle

For any single particle of mass mim_i moving with velocity v⃗i\vec{v}_i, its linear momentum is defined as p⃗i=miv⃗i\vec{p}_i = m_i \vec{v}_i. This is a vector quantity pointing in the same direction as the velocity. The SI unit of momentum is the kilogram-metre per second (kg⋅m/s\text{kg}\cdot\text{m/s}).

Summing Over All Particles

For a system of NN particles, the total linear momentum P⃗\vec{P} is the vector sum of all individual momenta: \vec{P} = \sum_{i=1}^{N} \vec{p}_i = \sum_{i=1}^{N} m_i \vec{v}_i. Because vectors add component-by-component, you can write Px=∑mivixP_x = \sum m_i v_{ix}, Py=∑miviyP_y = \sum m_i v_{iy}, and Pz=∑mivizP_z = \sum m_i v_{iz} separately.

Connecting to the Center of Mass

The position of the center of mass is defined as r⃗cm=1M∑i=1Nmir⃗i\vec{r}_{cm} = \frac{1}{M}\sum_{i=1}^{N} m_i \vec{r}_i, where M=∑miM = \sum m_i is the total mass. Differentiating with respect to time gives the velocity of the center of mass: v⃗cm=dr⃗cmdt=1M∑i=1Nmiv⃗i\vec{v}_{cm} = \frac{d\vec{r}_{cm}}{dt} = \frac{1}{M}\sum_{i=1}^{N} m_i \vec{v}_i. Comparing this with the expression for P⃗\vec{P} immediately yields the elegant result P⃗=Mv⃗cm\vec{P} = M\vec{v}_{cm}. The total momentum of the system equals the total mass multiplied by the velocity of the center of mass — exactly as if all the mass were concentrated at one point moving with velocity v⃗cm\vec{v}_{cm}.

Newton's Second Law for Systems

Taking the time derivative of the total momentum: dP⃗dt=∑i=1Nmia⃗i\frac{d\vec{P}}{dt} = \sum_{i=1}^{N} m_i \vec{a}_i. By Newton's second law applied to each particle, mia⃗i=F⃗iext+F⃗iintm_i \vec{a}_i = \vec{F}_i^{\text{ext}} + \vec{F}_i^{\text{int}}, where F⃗iext\vec{F}_i^{\text{ext}} is the net external force on particle ii and F⃗iint\vec{F}_i^{\text{int}} is the total internal force from all other particles. Newton's third law guarantees that every internal force pair sums to zero: ∑iF⃗iint=0⃗\sum_i \vec{F}_i^{\text{int}} = \vec{0}. Therefore, dP⃗dt=F⃗netext\frac{d\vec{P}}{dt} = \vec{F}_{\text{net}}^{\text{ext}}. Only external forces can change the system's total momentum.

Conservation of Momentum

When the net external force on a system is zero (F⃗netext=0⃗\vec{F}_{\text{net}}^{\text{ext}} = \vec{0}), it follows directly that dP⃗dt=0\frac{d\vec{P}}{dt} = 0, which means P⃗=constant\vec{P} = \text{constant}. This is conservation of linear momentum. It holds independently in each coordinate direction: if no external force acts in the xx-direction, then PxP_x is conserved even if PyP_y is not. This component-by-component conservation is extremely useful when analyzing oblique collisions or explosions.

The Key Equation

P⃗=∑i=1Nmiv⃗i=Mv⃗cm\vec{P} = \sum_{i=1}^{N} m_i \vec{v}_i = M\vec{v}_{cm}
  • \vec{P} — Total linear momentum (kg·m/s): The vector sum of all individual particle momenta in the system.
  • m_i — Mass of particle i (kg): The mass of the i-th particle in the system.
  • \vec{v}_i — Velocity of particle i (m/s): The velocity vector of the i-th particle.
  • M — Total mass (kg): The scalar sum of all particle masses: M=∑i=1NmiM = \sum_{i=1}^{N} m_i.
  • \vec{v}_{cm} — Center-of-mass velocity (m/s): The velocity of the center of mass of the entire system.

Worked Example

Problem. Three ice hockey pucks slide frictionlessly on a flat ice surface. Puck A has mass mA=0.20 kgm_A = 0.20\text{ kg} and velocity v⃗A=(3.0i^+0j^) m/s\vec{v}_A = (3.0\hat{i} + 0\hat{j})\text{ m/s}. Puck B has mass mB=0.20 kgm_B = 0.20\text{ kg} and velocity v⃗B=(−1.5i^+2.0j^) m/s\vec{v}_B = (-1.5\hat{i} + 2.0\hat{j})\text{ m/s}. Puck C has mass mC=0.40 kgm_C = 0.40\text{ kg} and velocity v⃗C=(0i^−1.0j^) m/s\vec{v}_C = (0\hat{i} - 1.0\hat{j})\text{ m/s}. (a) Find the total linear momentum of the system. (b) Find the speed of the center of mass.

P⃗=0.30i^kg ⋅ m/s,∣v⃗cm∣=0.375 m/s\vec{P} = 0.30\hat{i}\text{kg}\,\cdot\,\text{m/s},\quad |\vec{v}_{cm}| = 0.375\text{ m/s}

Answer. Total momentum is 0.30 kg·m/s in the positive x-direction; center-of-mass speed is 0.375 m/s.

Figure
Diagram
Total Linear Momentum of a Three-Particle Systemm₁p⃗₁m₂p⃗₂m₃p⃗₃CMP⃗= Mv⃗cmP⃗ = Σmᵢv⃗ᵢ = p⃗₁ + p⃗₂ + p⃗₃ = Mv⃗cmIndividual momenta (red arrows) add as vectors.The orange CM point moves with velocity v⃗cm = P⃗ / M.
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton's three laws of motion, published in his landmark 1687 work Philosophiæ Naturalis Principia Mathematica, provide the foundation for the concept of linear momentum. Newton defined 'quantity of motion' (what we now call momentum) as the product of mass and velocity, and his second law states that the rate of change of this quantity equals the net force. His third law directly implies that internal forces cannot change a system's total momentum — the core insight behind momentum conservation. Newton wrote the Principia in about 18 months, reportedly after astronomer Edmond Halley visited him and offered to pay the publication costs himself when he realized what Newton had discovered.

  • Center of Mass — The total momentum of a system equals total mass times center-of-mass velocity (P⃗=Mv⃗cm\vec{P} = M\vec{v}_{cm}), making the center of mass essential for understanding system momentum.
  • Conservation of Linear Momentum — When net external force is zero, total system momentum is conserved — a direct consequence of the system momentum framework developed here.
  • Newton's Third Law — The cancellation of internal force pairs, guaranteed by Newton's third law, is exactly why internal forces cannot change a system's total momentum.
  • Impulse–Momentum Theorem — The impulse–momentum theorem J⃗=ΔP⃗\vec{J} = \Delta\vec{P} extends system momentum ideas to situations involving time-varying external forces.
  • Angular Momentum of Particle Systems — Angular momentum of a system is the rotational analogue of linear momentum and obeys a parallel set of conservation laws, central to the Rotational Motion unit.