Momentum and collisions

Some events are over almost before they start: a hammer striking a nail, a car hitting a barrier, two railcars coupling, a jet of water hitting a turbine blade. The forces involved are large, short-lived and hard to measure, so Newton's second law in its \(F = ma\) form is awkward to use. Momentum handles these situations elegantly. It also leads to a conservation law that holds even when energy is being lost to heat and deformation.

This chapter defines momentum and impulse, derives the conservation of momentum from Newton's third law, classifies collisions, and extends the ideas to two dimensions, to the center of mass and to steady flows such as jets and rockets.

Momentum

The linear momentum of an object is its mass times its velocity:

Momentum is a vector, in the direction of the velocity, with units of kg·m/s (equivalently N·s). A 20 t railcar at 3 m/s and a 60 kg sprinter at 1000 m/s (if such a thing were possible) would have the same momentum. Momentum measures how hard it is to stop something, in the sense of how much push for how long.

Newton actually stated his second law in terms of momentum: the net force equals the rate of change of momentum,

For constant mass this is the same as \(m\mathbf{a}\). The momentum form is more general: it also covers systems whose mass changes, such as rockets and conveyor belts being loaded.

Impulse

Integrating the second law over the time a force acts gives the impulse–momentum theorem:

The impulse \(\mathbf{J}\) is the area under the force–time graph. During a collision the force rises and falls in a complicated way, but you rarely need its detailed shape. The change in momentum tells you the impulse, and dividing by the contact time gives the average force: \(F_{\text{avg}} = \Delta p/\Delta t\).

The same impulse can be delivered by a large force for a short time or a small force for a long time. Every piece of protective equipment, from airbags and helmets to crumple zones, packaging foam and the give in a climbing rope, works by stretching out the stopping time to reduce the force.

Conservation of momentum

Consider two objects that interact, say two carts colliding. By the third law, the force of A on B is equal and opposite to the force of B on A at every instant, and the two forces act for the same time. Their impulses are therefore equal and opposite, so whatever momentum A gains, B loses. The total is unchanged:

More generally: if the net external force on a system is zero, the total momentum of the system is constant. Internal forces, however large, only move momentum around within the system.

In collisions this holds to an excellent approximation even when external forces such as friction or gravity are present, because the collision forces are so much larger and act so briefly that external impulses during the impact are negligible. So for any collision or explosion, momentum just before equals momentum just after.

The same principle drives jet engines and rockets, explains why a fire hose kicks back and why a person stepping off a small boat pushes it away from the dock.

Types of collisions

Collisions are classified by what happens to kinetic energy.

• Elastic: kinetic energy is conserved. Nearly true for steel ball bearings, billiard balls, and gas molecules.
• Inelastic: some kinetic energy becomes heat, sound and deformation. Most real collisions.
• Perfectly inelastic: the objects stick together and move off with a common velocity. This loses the most kinetic energy that momentum conservation allows.

The coefficient of restitution \(e\) measures where a collision falls on this scale: the ratio of the speed of separation to the speed of approach, from \(e = 1\) (elastic) to \(e = 0\) (perfectly inelastic).

For a head-on elastic collision, momentum and energy conservation together give two equations for the two final velocities. With object B initially at rest, the solution is

Equal masses swap velocities (the moving one stops dead, as in a Newton's cradle). A light object bounces off a heavy one; a heavy object barely slows when it hits a light one.

Collisions in two dimensions

Momentum is a vector, so in two dimensions it is conserved separately in each direction: \(\sum p_x\) before equals \(\sum p_x\) after, and likewise for \(y\). That gives two equations, which is enough for a perfectly inelastic collision (two unknown velocity components).

Collision investigators run this calculation backward. From the masses, the direction the wreckage traveled (from tire marks) and the distance it slid (from the work done by friction, Lesson 3), they can work out each vehicle's speed before impact.

The center of mass

For a system of particles, the center of mass is the mass-weighted average position:

and similarly for \(y\) and \(z\). (For continuous bodies, the sums become the integrals you met in Calculus II.) The total momentum of a system equals its total mass times the velocity of its center of mass, so

The center of mass moves as if the whole mass were concentrated there and all the external forces acted on it. Internal forces cannot change its motion. A shell that explodes in mid-flight sends fragments everywhere, but their center of mass continues along the original parabola. A person walking to the front of a floating canoe moves the canoe backward so that the center of mass of person plus canoe stays put.

Momentum in flowing systems

For a steady stream of fluid, the momentum form of the second law gives the force needed to change the stream's velocity: force = (mass flow rate) × (change in velocity), \(F = \dot m\,\Delta v\). This is the basis of the momentum equation in fluid mechanics.

Rockets and jet engines work the same way in reverse: they throw mass backward at high speed, and the thrust is the mass flow rate times the exhaust velocity. A rocket burning 200 kg/s with an exhaust velocity of 3000 m/s produces 600 kN of thrust.