1.10 - Momentum Conservation

Introduction

The conservation of momentum is one of the most fundamental and powerful principles in physics. It states that the total momentum of an isolated system of particles remains constant over time if the system is not subject to any external forces. This principle follows directly from Newton's Third Law and provides a powerful tool for analyzing collisions, explosions, and other interactions between objects. Understanding momentum conservation is essential for solving many problems in classical mechanics.

Momentum of a Single Particle

Before discussing systems of particles, we recall that the momentum of a single particle is defined as:

Momentum of a Single Particle:

\mathbf{p} = m\mathbf{v}

where:

$\mathbf{p}$ is the momentum vector
$m$ is the mass of the particle
$\mathbf{v}$ is the velocity vector of the particle

Momentum is a vector quantity with both magnitude and direction. Its SI units are $\text{kg} \cdot \text{m/s}$.

Total Momentum for a System of Particles

For a system consisting of $N$ particles, the total momentum of the system is defined as the vector sum of the individual momenta of all particles in the system.

Total Momentum of a System:

For a system of $N$ particles with masses $m_1, m_2, \ldots, m_N$ and velocities $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_N$, the total momentum is:

\mathbf{P}_\text{total} = \sum_{i=1}^N \mathbf{p}_i = \sum_{i=1}^N m_i \mathbf{v}_i = m_1\mathbf{v}_1 + m_2\mathbf{v}_2 + \cdots + m_N\mathbf{v}_N

where $\mathbf{p}_i = m_i\mathbf{v}_i$ is the momentum of the $i$-th particle.

Properties of Total Momentum:

Vector quantity: Total momentum is a vector, so it has both magnitude and direction. When adding momenta, we must add them as vectors, not just magnitudes.
Component form: In Cartesian coordinates, the total momentum can be written as: $$\mathbf{P}_\text{total} = P_x\hat{x} + P_y\hat{y} + P_z\hat{z}$$ where $P_x = \sum_{i=1}^N m_i v_{ix}$, and similarly for $P_y$ and $P_z$.
Independent of internal forces: The total momentum depends only on the masses and velocities of the particles, not on the forces between them.

Derivation: Momentum Conservation from Newton's Third Law

We now show that momentum conservation follows directly from Newton's Third Law. This is a fundamental result that connects Newton's laws to one of the most important conservation principles in physics.

Step 1: Newton's Second Law for Each Particle

Consider a system of $N$ particles. For each particle $i$, Newton's Second Law states:

\mathbf{F}_{i,\text{net}} = \frac{d\mathbf{p}_i}{dt} = \frac{d(m_i\mathbf{v}_i)}{dt}

where $\mathbf{F}_{i,\text{net}}$ is the net force acting on particle $i$. This net force can be written as the sum of:

External forces: Forces from objects outside the system acting on particle $i$, denoted $\mathbf{F}_{i,\text{ext}}$
Internal forces: Forces from other particles within the system acting on particle $i$, denoted $\mathbf{F}_{ij}$ (force on particle $i$ due to particle $j$)

Therefore:

\mathbf{F}_{i,\text{net}} = \mathbf{F}_{i,\text{ext}} + \sum_{j \neq i} \mathbf{F}_{ij} = \frac{d\mathbf{p}_i}{dt}

where the sum is over all other particles $j$ in the system.

Step 2: Sum Over All Particles

To find the rate of change of the total momentum, we sum the above equation over all particles:

\sum_{i=1}^N \mathbf{F}_{i,\text{net}} = \sum_{i=1}^N \mathbf{F}_{i,\text{ext}} + \sum_{i=1}^N \sum_{j \neq i} \mathbf{F}_{ij} = \sum_{i=1}^N \frac{d\mathbf{p}_i}{dt}

The right-hand side is the time derivative of the total momentum:

\sum_{i=1}^N \frac{d\mathbf{p}_i}{dt} = \frac{d}{dt}\left(\sum_{i=1}^N \mathbf{p}_i\right) = \frac{d\mathbf{P}_\text{total}}{dt}

So we have:

\sum_{i=1}^N \mathbf{F}_{i,\text{ext}} + \sum_{i=1}^N \sum_{j \neq i} \mathbf{F}_{ij} = \frac{d\mathbf{P}_\text{total}}{dt}

Step 3: Apply Newton's Third Law

Now we apply Newton's Third Law. For any pair of particles $i$ and $j$:

\mathbf{F}_{ij} = -\mathbf{F}_{ji}

This means the force on particle $i$ due to particle $j$ is equal and opposite to the force on particle $j$ due to particle $i$.

When we sum over all internal forces:

\sum_{i=1}^N \sum_{j \neq i} \mathbf{F}_{ij} = \sum_{\text{all pairs } (i,j)} (\mathbf{F}_{ij} + \mathbf{F}_{ji})

For each pair $(i,j)$, we have $\mathbf{F}_{ij} + \mathbf{F}_{ji} = \mathbf{F}_{ij} - \mathbf{F}_{ij} = \mathbf{0}$ by Newton's Third Law. Therefore:

\sum_{i=1}^N \sum_{j \neq i} \mathbf{F}_{ij} = \mathbf{0}

This is the key result: The sum of all internal forces in a system is zero because they occur in equal and opposite pairs.

Step 4: The Conservation Law

Substituting this result into our equation:

\sum_{i=1}^N \mathbf{F}_{i,\text{ext}} + \mathbf{0} = \frac{d\mathbf{P}_\text{total}}{dt}

which simplifies to:

Rate of Change of Total Momentum:

\sum_{i=1}^N \mathbf{F}_{i,\text{ext}} = \frac{d\mathbf{P}_\text{total}}{dt}

The rate of change of the total momentum of a system equals the sum of all external forces acting on the system. This is Newton's Second Law for the system as a whole.

Step 5: Momentum Conservation for Isolated Systems

If the system is isolated (no external forces act on it), then:

\sum_{i=1}^N \mathbf{F}_{i,\text{ext}} = \mathbf{0}

Therefore:

\frac{d\mathbf{P}_\text{total}}{dt} = \mathbf{0}

This means the total momentum is constant:

Conservation of Momentum:

For an isolated system of particles (no external forces), the total momentum is conserved:

\mathbf{P}_\text{total} = \text{constant}

or equivalently:

\mathbf{P}_\text{total}(t_1) = \mathbf{P}_\text{total}(t_2)

for any times $t_1$ and $t_2$.

Key Points in the Derivation:

We started with Newton's Second Law applied to each particle.
We summed over all particles to get the rate of change of total momentum.
Newton's Third Law caused all internal forces to cancel: This is the crucial step. Because internal forces occur in action-reaction pairs ($\mathbf{F}_{ij} = -\mathbf{F}_{ji}$), their sum is zero.
For an isolated system (no external forces), the total momentum is constant.

Momentum Conservation in Component Form

Since momentum is a vector, conservation of momentum applies to each component separately. For an isolated system:

P_x = \sum_{i=1}^N m_i v_{ix} = \text{constant}

P_y = \sum_{i=1}^N m_i v_{iy} = \text{constant}

P_z = \sum_{i=1}^N m_i v_{iz} = \text{constant}

This means that even if external forces act in some directions but not others, momentum is conserved in the directions where there are no external forces.

Examples and Applications

Example 1: Two-Particle Collision

Two particles with masses $m_1$ and $m_2$ collide. Before the collision, particle 1 has velocity $\mathbf{v}_{1i}$ and particle 2 has velocity $\mathbf{v}_{2i}$. After the collision, they have velocities $\mathbf{v}_{1f}$ and $\mathbf{v}_{2f}$.

If no external forces act on the system, momentum is conserved:

m_1\mathbf{v}_{1i} + m_2\mathbf{v}_{2i} = m_1\mathbf{v}_{1f} + m_2\mathbf{v}_{2f}

This equation holds even though the individual velocities change. The internal forces during the collision (which are action-reaction pairs) cancel out, leaving the total momentum unchanged.

Example 2: Explosion

A firework of mass $M$ initially at rest explodes into three pieces with masses $m_1$, $m_2$, and $m_3$, moving with velocities $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{v}_3$.

Since the firework was initially at rest, the initial total momentum is zero. By momentum conservation:

\mathbf{0} = m_1\mathbf{v}_1 + m_2\mathbf{v}_2 + m_3\mathbf{v}_3

This means the three momenta must sum to zero. The pieces fly apart in such a way that their total momentum is zero.

Example 3: Rocket Propulsion

A rocket of mass $M$ ejects fuel of mass $dm$ with velocity $\mathbf{v}_\text{exhaust}$ relative to the rocket. If the rocket is in space (no external forces), momentum is conserved.

Initially, the rocket and fuel have total momentum $\mathbf{P}_i$. After ejection, the rocket (now mass $M - dm$) has velocity $\mathbf{v}_\text{rocket}$ and the fuel has velocity $\mathbf{v}_\text{rocket} + \mathbf{v}_\text{exhaust}$.

Momentum conservation gives:

\mathbf{P}_i = (M - dm)\mathbf{v}_\text{rocket} + dm(\mathbf{v}_\text{rocket} + \mathbf{v}_\text{exhaust})

This shows how the rocket gains forward momentum by ejecting fuel backward—a direct application of momentum conservation and Newton's Third Law.

Example 4: Two Ice Skaters

Two ice skaters initially at rest push against each other. Skater 1 (mass $m_1$) pushes on Skater 2, and Skater 2 (mass $m_2$) pushes back on Skater 1. They move apart with velocities $\mathbf{v}_1$ and $\mathbf{v}_2$.

Since they started at rest, initial momentum is zero. By momentum conservation:

\mathbf{0} = m_1\mathbf{v}_1 + m_2\mathbf{v}_2

Therefore, $m_1\mathbf{v}_1 = -m_2\mathbf{v}_2$. The skaters move in opposite directions, and the lighter skater moves faster (has greater speed) than the heavier one.

Systems with External Forces

When external forces act on a system, the total momentum is not conserved. However, the rate of change of total momentum equals the sum of external forces:

\frac{d\mathbf{P}_\text{total}}{dt} = \sum_{i=1}^N \mathbf{F}_{i,\text{ext}}

This is still a very useful result. For example:

If external forces are zero in a particular direction (say, the $x$-direction), then $P_x$ is conserved even if other components are not.
If the sum of external forces is constant, the total momentum changes at a constant rate.
The center of mass of a system moves as if all the mass were concentrated there and all external forces acted on it.

Connection to Center of Mass

The total momentum is related to the velocity of the center of mass. The center of mass position is:

\mathbf{R}_\text{CM} = \frac{\sum_{i=1}^N m_i \mathbf{r}_i}{\sum_{i=1}^N m_i} = \frac{\sum_{i=1}^N m_i \mathbf{r}_i}{M_\text{total}}

where $M_\text{total} = \sum_{i=1}^N m_i$ is the total mass. The center of mass velocity is:

\mathbf{V}_\text{CM} = \frac{d\mathbf{R}_\text{CM}}{dt} = \frac{\sum_{i=1}^N m_i \mathbf{v}_i}{M_\text{total}} = \frac{\mathbf{P}_\text{total}}{M_\text{total}}

Therefore:

\mathbf{P}_\text{total} = M_\text{total} \mathbf{V}_\text{CM}

This shows that the total momentum equals the total mass times the center of mass velocity. For an isolated system, since $\mathbf{P}_\text{total}$ is constant, the center of mass moves with constant velocity.

Summary

Key Takeaways:

The total momentum of a system of $N$ particles is $\mathbf{P}_\text{total} = \sum_{i=1}^N m_i \mathbf{v}_i$.
Momentum conservation follows from Newton's Third Law: Internal forces occur in action-reaction pairs that cancel when summed, leaving only external forces to change the total momentum.
For an isolated system (no external forces), the total momentum is conserved: $\mathbf{P}_\text{total} = \text{constant}$.
The rate of change of total momentum equals the sum of external forces: $\frac{d\mathbf{P}_\text{total}}{dt} = \sum_{i=1}^N \mathbf{F}_{i,\text{ext}}$.
Momentum conservation is a powerful tool for analyzing collisions, explosions, and other interactions where internal forces dominate.
The total momentum equals the total mass times the center of mass velocity: $\mathbf{P}_\text{total} = M_\text{total} \mathbf{V}_\text{CM}$.
Momentum conservation applies to each component separately. If there are no external forces in a particular direction, momentum is conserved in that direction.