Momentum and collisions

Why this matters

Collisions happen too quickly, and with forces too complicated, to track with F = m·a moment by moment. Momentum lets you ignore what happens during the crash and still predict what comes out of it. Crash reconstruction, the design of airbags and crumple zones, rocket propulsion, particle physics and even pool shots all rest on one fact: in an isolated system, total momentum does not change.

Momentum

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

p = m · v        (units: kg·m/s)

Momentum is a vector: it points the same way as the velocity. In one dimension, a sign is enough. A 1,000 kg car moving east at 20 m/s has p = +20,000 kg·m/s; the same car moving west has −20,000 kg·m/s.

Impulse

A force acting for a time changes momentum. The product is called impulse:

J = F · Δt = Δp

(For a force that changes during the impact, F is the average force.) The same change in momentum can come from a big force for a short time or a smaller force for a longer time.

A 0.145 kg baseball arrives at 40 m/s and is stopped by a glove in 0.020 s:

Δp = 0.145 × 40 = 5.8 kg·m/s
F  = 5.8 / 0.020 = 290 N

If the catcher lets the glove move back so the stop takes 0.040 s, the average force halves to 145 N. Airbags, crumple zones, helmet foam and bending your knees when you land all work this way: they stretch out Δt to cut the force.

Conservation of momentum

When two objects collide, Newton's third law says they push on each other with equal and opposite forces for the same time. So the impulse on one is exactly the opposite of the impulse on the other, and the total momentum is unchanged:

m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'

(Primes mean "after the collision".) This holds as long as outside forces such as friction are small during the brief collision, which is usually a good approximation.

Types of collisions

Momentum is always conserved in a collision. Kinetic energy is not always conserved:

• Elastic: kinetic energy is conserved too. Close to true for billiard balls and for collisions between atoms.
• Inelastic: some kinetic energy turns into heat, sound and deformation.
• Perfectly inelastic: the objects stick together and move off as one. This loses the most kinetic energy that momentum conservation allows.

Worked example 1: two cars lock together

A 1,000 kg car moving at 20 m/s rear-ends a 1,500 kg car stopped at a light. The bumpers lock. How fast do they move just after impact?

before:  1000 × 20 + 1500 × 0 = 20,000 kg·m/s
after:   (1000 + 1500) × v'   = 20,000
         v' = 20,000 / 2,500  = 8.0 m/s

Kinetic energy check:

before:  ½ × 1000 × 20²  = 200,000 J
after:   ½ × 2500 × 8.0² =  80,000 J

60% of the kinetic energy, 120 kJ, went into crumpling metal, heat and sound. Crash reconstruction works backwards from skid marks and final positions using exactly these two ideas, plus a lot of careful measurement.

Worked example 2: an elastic collision

For a one-dimensional elastic collision, combining momentum and kinetic-energy conservation gives:

v₁' = ((m₁ − m₂)·v₁ + 2·m₂·v₂) / (m₁ + m₂)
v₂' = ((m₂ − m₁)·v₂ + 2·m₁·v₁) / (m₁ + m₂)

Two equal billiard balls (m₁ = m₂), the first moving at 2.0 m/s and the second at rest: v₁' = 0 and v₂' = 2.0 m/s. The moving ball stops dead and the other takes all its velocity, which is exactly what you see in a straight pool shot.

In code

def total_momentum(masses, velocities):
    return sum(m * v for m, v in zip(masses, velocities))

print(total_momentum([1000, 1500], [20, 0]))   # 20000

Writing the before and after totals as numbers you can compare is a good habit: if they differ, you have made a sign or arithmetic error.

Common mistakes

• Dropping the signs. Objects moving in opposite directions have momenta of opposite sign. A head-on collision of equal cars at equal speeds has zero total momentum.
• Assuming kinetic energy is conserved. Only in elastic collisions. Momentum is the reliable one.
• Using conservation when a big outside force acts. A car hitting a wall is not isolated: the wall (and the Earth) take the momentum.
• Confusing momentum and kinetic energy. p = mv is a vector and grows with v; KE = ½mv² is a scalar and grows with v².

Quick recap

• p = m·v; impulse F·Δt = Δp.
• Longer stopping time means smaller average force.
• In a collision, total momentum before equals total momentum after.
• Perfectly inelastic: v' = (m₁v₁ + m₂v₂) / (m₁ + m₂).
• Kinetic energy is conserved only in elastic collisions.

Try it

In the Do tab, write the function for a perfectly inelastic collision and use it to measure how much kinetic energy is lost.