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Physics C: Mechanics Unit 4: Systems of Particles and Linear Momentum

Momentum lets you handle collisions, explosions and systems of many particles without knowing the messy internal forces. This unit covers impulse, conservation of momentum, collisions and center-of-mass motion.

Covers: Center of mass of discrete and continuous systems, Impulse and the impulse–momentum theorem, Conservation of linear momentum, Elastic and inelastic collisions, Two-dimensional collisions, Center-of-mass motion and explosions, Variable mass and rocket propulsion

Study guide

Linear momentum is p = mv, a vector. Newton's second law in its general form is F = dp/dt, so the impulse delivered by a force, J = ∫F dt, equals the change in momentum. On a force–time graph, impulse is the area under the curve, and the average force over a contact time is Δp/Δt. This explains airbags and padded gloves: spreading the same momentum change over a longer time lowers the average force. Remember that momentum is a vector, so a ball that reverses direction has a change in momentum equal to its mass times the sum of the two speeds.

For a system of particles, the center of mass is the mass-weighted average position, x_cm = Σmᵢxᵢ/M. For a continuous object, x_cm = (1/M)∫x dm, where dm = λ dx for a rod with linear density λ. The total momentum equals M v_cm, and the center of mass accelerates as if all the external forces acted on a single particle of mass M: ΣF_ext = M a_cm. Internal forces, such as those in an explosion or between a person and a boat, cannot move the center of mass. That makes many problems quick: the pieces of an exploding projectile have a center of mass that stays on the original parabola.

If the net external force on a system is zero, its total momentum is conserved. Collisions are short enough that outside forces like gravity or friction deliver little impulse during the impact, so momentum is conserved across the collision even when external forces act before and after. In two dimensions, conserve each component separately. Momentum conservation always holds in isolated collisions; kinetic energy may or may not be conserved, and that is what classifies a collision.

In an elastic collision, kinetic energy is also conserved. In one dimension with a target at rest, v₁′ = (m₁ − m₂)v₁/(m₁ + m₂) and v₂′ = 2m₁v₁/(m₁ + m₂). Equal masses swap velocities, and in a glancing elastic collision between equal masses the final velocities are perpendicular. A useful shortcut is that each object's velocity reflects about v_cm: v′ = 2v_cm − v. In an inelastic collision, some kinetic energy becomes heat or deformation. A perfectly inelastic collision, where objects stick, loses the most kinetic energy. The coefficient of restitution e compares relative speeds after and before; e = 1 is elastic and e = 0 is perfectly inelastic.

When mass enters or leaves a system, use momentum rather than F = ma. A rocket that ejects exhaust at speed v_ex relative to itself feels a thrust v_ex(dm/dt), and in deep space its speed changes by Δv = v_ex ln(m₀/m_f). Similar thinking applies to sand dropped onto a moving belt: the belt must supply a force v(dm/dt) to bring the sand up to speed, and the power required is twice the rate at which the sand gains kinetic energy, because sliding friction dissipates the rest. Ballistic pendulum problems split into two stages: momentum conservation in the collision and energy conservation in the swing afterward.

Key ideas

• Impulse ∫F dt equals the change in momentum.
• Momentum is conserved when the net external force is zero.
• Internal forces cannot change the motion of the center of mass.
• All isolated collisions conserve momentum, but only elastic ones conserve kinetic energy.
• Perfectly inelastic collisions lose the most kinetic energy.
• Two-dimensional momentum problems conserve each component separately.
• Variable-mass systems need the momentum form of Newton's second law.

Formulas and rules

Common mistakes

• Forgetting that momentum is a vector and subtracting speeds when an object reverses.
• Using conservation of kinetic energy in an inelastic collision.
• Applying energy conservation across the embedding step of a ballistic pendulum.
• Using the plain average of positions instead of the mass-weighted center of mass.
• Adding momentum components in two dimensions as if they were parallel.

Before the test, make sure you can

• Calculate impulse from a force function or force–time graph, and find average force.
• Locate the center of mass of point masses and of a rod with variable density.
• Apply momentum conservation to collisions and explosions in one and two dimensions.
• Classify collisions and calculate kinetic energy lost.
• Use the elastic collision formulas and the center-of-mass shortcut.
• Solve ballistic pendulum problems in two stages.
• Use center-of-mass reasoning for explosions and people walking on boats.
• Calculate rocket thrust and Δv.

Words to know

Linear momentum

The product of mass and velocity, p = mv; a vector.

Impulse

The time integral of force, J = ∫F dt, equal to the change in momentum.

Impulse–momentum theorem

The net impulse on an object equals its change in momentum.

Center of mass

The mass-weighted average position of a system.

Isolated system

A system with zero net external force, so its momentum is conserved.

Internal force

A force between parts of the system; internal forces cancel in pairs.

Elastic collision

A collision that conserves both momentum and kinetic energy.

Inelastic collision

A collision that conserves momentum but not kinetic energy.

Perfectly inelastic collision

A collision in which the objects stick together and move as one.

Coefficient of restitution

The ratio of relative speed after a collision to relative speed before.

Ballistic pendulum

A device that measures a projectile's speed from the swing of a block it embeds in.

Thrust

The forward force on a rocket, equal to exhaust speed times mass flow rate.

Rocket equation

Δv = v_ex ln(m₀/m_f), the speed gain of a rocket in deep space.

Linear mass density

Mass per unit length, λ = dm/dx.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.