Momentum

Momentum, and the law that defines how it changes

Linear momentum is the product of a body's mass and its velocity, p = m v. Because velocity is a vector, momentum is a vector: it has a direction, and in one dimension that direction shows up as a sign you are not allowed to drop. Its SI unit is the kilogram meter per second, kg*m/s. A 1,200 kg car heading east at 20 m/s and an identical car heading west at 20 m/s have momenta of equal magnitude and opposite sign: together they carry zero net momentum while carrying a great deal of kinetic energy.

Force and Motion already introduced the general form of Newton's second law: the net external force on a body equals the rate of change of its momentum, F_net = dp/dt. Momentum is the quantity that statement is about, and the familiar F_net = m a is what you get by assuming mass is constant and pulling it outside the derivative. Everything here follows from taking dp/dt seriously: impulse is what you get by integrating it over time, conservation is what you get when the left side is zero, and rocket propulsion is what you get when m is not constant.

Impulse: force accumulated over time

Integrate the second law over an interval and the derivative disappears. The impulse delivered by a force is its time integral, J = integral of F(t) dt over the interval, and the result is the impulse-momentum theorem: J = delta p. The impulse applied to a body equals the change in that body's momentum, exactly, with no approximation and no assumption that the force was steady.

The units make the same point. A newton is one kgms^-2, so a newton second is kgms^-2 * s = kg*m/s. Impulse and momentum are measured in the same unit because they are the same kind of quantity, viewed once as a cause and once as an effect.

In a real impact the force history F(t) is a spike that rises and falls over a few tens of milliseconds, and nobody knows its exact shape without instrumenting the event. The theorem still works, because the integral is fixed by the endpoints. If you know the momentum before and after, you know the total impulse regardless of what happened in between, and dividing that impulse by the duration gives the average force, F_ave = J / delta t. The average force is a real, useful number, but it is not the peak force. A short, sharp force history can peak well above its own average, which is why crash engineering is measured rather than inferred from averages.

Why stretching the collision reduces the force

Here is the engineering payoff of the whole topic. A body that must be brought from some speed to rest has a change in momentum, delta p = m * delta v, set by its mass and its speed. That is fixed by the situation. How long the stopping takes is not.

Since J = delta p and F_ave = J / delta t, holding delta p constant while increasing delta t drives the average force down in exact inverse proportion. Double the stopping time and you halve the average force. Nothing absorbed the momentum, because momentum is not the kind of thing you absorb; it was simply removed more gradually.

That single relationship is the physical principle underneath crumple zones, airbags, helmet liners, fall-arrest lanyards, running-track surfaces, and the foam in a shipping carton. Each of them is a device for making delta t larger. A crumple zone lengthens the time over which the vehicle structure decelerates; an airbag and a belt lengthen the time over which the occupant decelerates, which is a separate and later event; a crushable helmet liner lengthens the time over which the skull decelerates. In every case the momentum change is whatever the crash makes it, and the design variable is time.

Conservation belongs to the system, not to the body

Return to F_net = dp/dt and set the left-hand side to zero. If the net external force on a system is zero, the system's total momentum does not change with time. That is conservation of linear momentum, and it is a consequence of the second law rather than an independent postulate.

Two pieces of precision matter more than the statement itself. The first is the word system. A system is whatever collection of bodies you decide to draw a boundary around, and forces are internal or external only relative to that boundary. Forces internal to the system come in third-law pairs, so their contributions to the total momentum change cancel identically; only external forces can move the total. Choosing the boundary well is the skill. Put both colliding cars inside it and the enormous contact forces become internal and drop out. Put one car inside it and the same forces are external and you have gained nothing.

The second is that the conserved quantity is the total. Each body's momentum changes, often dramatically. What is constant is the vector sum. Saying momentum is conserved in a collision does not mean any particular object kept its momentum; it means the momentum one object lost, another gained.

External forces are rarely exactly zero. Gravity and friction act on colliding cars the whole time. The usual justification is that a collision is brief and the contact forces are enormous, so over the few tens of milliseconds of contact the external impulse is negligible next to the internal one. Momentum conservation across a collision is therefore an excellent approximation with a stated reason, not an exact law of the scene. It is also directional: momentum can be conserved along one axis and not along another, and treating each axis separately is standard practice.

Collisions: elastic, inelastic, and the coefficient of restitution

Collisions are classified by what happens to kinetic energy, because momentum is conserved in all of them. In an elastic collision the total kinetic energy after equals the total before. In an inelastic collision some kinetic energy is converted into deformation, heat, sound, and permanent damage, so the final kinetic energy is positive but smaller. In a perfectly inelastic collision the bodies stick together and move off as one composite object, and the kinetic-energy loss is the largest that momentum conservation permits.

The coefficient of restitution, usually written e or epsilon, puts a number on that spectrum. For two bodies in a head-on collision it is the negative ratio of relative velocity after impact to relative velocity before: e = -(v_Af - v_Bf) / (v_Ai - v_Bi). It runs from 0 to 1. A value of 1 is the elastic limit, where the pair separates exactly as fast as it approached; a value of 0 is the perfectly inelastic limit, where the relative velocity after impact is zero and the bodies do not separate at all. Intermediate values describe everything real. Restitution is a property of the colliding pair and the conditions, not of a material alone.

A warning about vocabulary. Engineering dynamics texts and physics texts do not use these words the same way. Physics texts distinguish inelastic from perfectly inelastic; several engineering dynamics texts call the stick-together case simply inelastic and use semi-elastic for everything in between. The physical categories are identical, so read whichever convention your course uses and check what the author means by inelastic before answering a problem.

Center of mass, and the motion of a system as a whole

A system of particles has a mass-weighted average position, the center of mass, r_CM = (1/M) * sum of m_j r_j, where M is the total mass. Differentiate it and two useful results appear. The total momentum of the system equals M v_CM, so the whole apparatus of conservation can be restated as a statement about how the center of mass moves. And the net external force equals M a_CM, meaning the center of mass accelerates exactly as a single particle of mass M would if all the external force were applied to it.

This is what licenses the habit of treating a car, a satellite, or a thrown wrench as a point. Internal forces, however violent, cannot move the center of mass off its trajectory. When a firework shell bursts, every fragment flies somewhere new, but the center of mass of the fragments continues along the same parabola the intact shell was on until air drag on the individual pieces, an external force, changes it.

Variable-mass systems and rocket propulsion

F_net = m a is a special case that fails whenever the system's mass changes while it moves. A rocket, a hopper being filled on a moving conveyor, and an aircraft burning fuel are all variable-mass problems, and the momentum form is the one that survives.

A rocket in free space carries no external force to push against, and it does not need one. It expels combustion products backwards at high speed; that exhaust carries momentum away, and since the total momentum of rocket-plus-exhaust cannot change, the rocket's momentum changes by the same magnitude in the opposite direction. Thrust is the rate at which momentum is carried off by the exhaust: the momentum-flux term is the exhaust speed multiplied by the mass flow rate, mdot * Ve. NASA's rocket thrust equation adds a second term for the pressure mismatch at the nozzle exit, F = mdot * Ve + (pe - p0) * Ae, which matters because the exit pressure equals the ambient pressure only at one design condition.

Two consequences are worth noticing. Constant thrust does not produce constant acceleration, because the mass in the denominator keeps falling as propellant burns. And integrating over the burn gives the rocket equation, delta v = u * ln(m_0/m), where u is the exhaust speed: velocity change depends on the logarithm of the mass ratio, which is why propellant mass grows punishingly fast with mission delta v.

Angular momentum, briefly

Rotation has its own conserved quantity. For a particle, angular momentum about a point is the cross product of the position vector with the linear momentum, l = r x p, measured in kg*m^2/s. For a rigid body rotating about a fixed axis the magnitude is L = I omega, with I the moment of inertia. The rotational analogue of the second law is that the net external torque about a point equals the rate of change of angular momentum about that point, so when the net external torque is zero, total angular momentum is conserved.

The spinning skater is the standard illustration and the standard place people get it wrong. Pulling the arms in moves mass closer to the rotation axis, which lowers I. With L = I omega fixed, omega must rise. The part usually mangled: rotational kinetic energy does not stay the same. Since K = L^2 / (2 I), halving I doubles it. That energy is not created by the conservation law; the skater does mechanical work pulling her arms inward, and that work is exactly what appears as extra kinetic energy. Angular momentum is conserved; energy is supplied.

Idealizations, and the scope of this lesson

Every result here rests on assumptions worth naming. Bodies were treated as particles or rigid bodies. The collision was treated as instantaneous relative to the external forces, which is why gravity and friction were dropped. Restitution was treated as a constant for the pair, when in practice it varies with speed, temperature, and geometry. Momentum conservation itself is exact; the modeling around it is not.

This lesson is educational material, not engineering design guidance. In particular, nothing here is guidance for designing occupant protection. Crumple zones, restraints, airbags, and helmets are governed by regulation and prescribed test procedure. In the United States, occupant crash protection for passenger vehicles falls under Federal Motor Vehicle Safety Standard No. 208 at 49 CFR 571.208, which sets performance requirements in terms of forces and accelerations measured on anthropomorphic dummies in specified test crashes, along with equipment requirements for active and passive restraint systems. Real design in this space requires a licensed engineer, verified material and component data, physical testing to the governing standard, and margins chosen for the consequences of failure. The arithmetic in this lesson explains why such systems work; it does not tell anyone how to build one.