8.2 Conservation of Linear Momentum

Conservation of Linear Momentum

Linear momentum describes how much "motion" an object carries, defined as the product of its mass and velocity. Conservation of momentum is one of the most powerful tools in physics because it lets you analyze collisions, explosions, and recoil problems even when you don't know the details of the forces involved.

Law of Conservation of Linear Momentum

Linear momentum is defined as $\vec{p} = m\vec{v}$, a vector quantity pointing in the same direction as the object's velocity.

The law of conservation of linear momentum states that the total momentum of a system remains constant as long as no net external force acts on it:

$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$

For a two-object system, this looks like:

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

Because momentum is a vector, it's conserved independently in each direction (x, y, z). In two-dimensional problems, you'll write separate conservation equations for each component.

Applications in Collisions and Explosions

Momentum conservation applies to every collision and explosion, but what differs is whether kinetic energy is also conserved.

• Elastic collisions conserve both momentum and kinetic energy. Billiard ball collisions are a close real-world approximation. You get two equations (one for momentum, one for kinetic energy), which lets you solve for two unknowns.
• Inelastic collisions conserve momentum but not kinetic energy. Some kinetic energy converts to heat, sound, or deformation. A car crash is a typical example.
• Perfectly inelastic collisions are the extreme case where the objects stick together after impact. Since they share a final velocity $\vec{v}_f$, the equation simplifies to $m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = (m_1 + m_2)\vec{v}_f$.
• Explosions work in reverse: an initially stationary object breaks apart, and the pieces fly off with momenta that sum to zero (since the system started with zero momentum).

Problem-solving steps:

1. Define your system and confirm that external forces are negligible (or that you're looking at a very short time interval).
2. Write out the total momentum before the event: $\vec{p}_{\text{initial}} = m_1\vec{v}_{1i} + m_2\vec{v}_{2i}$.
3. Write out the total momentum after the event: $\vec{p}_{\text{final}} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}$.
4. Set them equal and solve for the unknown(s). For 2D problems, do this separately for x and y components.

One more useful fact: the center of mass of a system moves at constant velocity regardless of what the internal forces do. Collisions and explosions rearrange how momentum is distributed among the pieces, but the center of mass keeps moving the same way.

Scenarios of Momentum Conservation

• Space collisions are the cleanest example because there's essentially no friction or air resistance, so external forces are truly zero.
• Gun recoil shows conservation clearly: a bullet with small mass flies forward at high speed, while the much heavier gun kicks backward at lower speed. Their momenta are equal in magnitude and opposite in direction, summing to zero (matching the initial state at rest).
• Rocket propulsion works by expelling exhaust gas backward. The momentum gained by the exhaust in one direction equals the momentum gained by the rocket in the other.
• Short-duration impacts like a bat hitting a baseball approximately conserve momentum because the collision happens so fast that external forces (gravity, friction) don't have time to deliver significant impulse during the event itself.
• Particle physics experiments rely heavily on momentum conservation to identify unknown particles produced in high-energy collisions.

Momentum Conservation and External Forces

Conservation of momentum isn't a separate law from Newton's laws; it follows directly from them.

Newton's Second Law in momentum form is $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$. When the net external force is zero, $\frac{d\vec{p}}{dt} = 0$, so momentum doesn't change. That's the conservation law.

Newton's Third Law is the reason internal forces don't matter. Every internal force has an equal and opposite partner within the system, so they cancel when you add up the total momentum.

The impulse-momentum theorem, $\Delta\vec{p} = \vec{F}_{\text{net}}\Delta t$, connects these ideas: if there is a net external force, it changes the system's momentum by an amount equal to the impulse. This is why choosing your system boundaries carefully matters. A force that's "external" for one choice of system might be "internal" for a larger system.

Finally, momentum conservation holds in all inertial reference frames (this is called Galilean invariance). You can solve a problem from any inertial frame and get consistent results, though some frames make the math easier than others.