Linear Momentum and Collisions

Big Picture

Momentum is "quantity of motion" — the product of mass and velocity. Like energy, momentum is conserved in isolated systems. But unlike energy, momentum is a vector and is conserved in each direction independently. Collisions are classified by whether kinetic energy is conserved (elastic) or not (inelastic).

ELI-10: Explain It Like I'm 10

Imagine two ice skaters pushing off each other. The heavier skater moves backward slowly; the lighter skater moves backward quickly. Their motion is balanced — the "total push" in the system is zero because when one pushes, the other gets an equal push in the opposite direction. That "total push" idea is momentum. In any collision or explosion, the total momentum before equals the total momentum after.

4.1 Momentum

Core Idea

Linear momentum is a vector quantity defined as the product of mass and velocity.

Physics and Mathematics

p = mv

Where:

• p = momentum (kg·m/s)
• m = mass (kg)
• v = velocity (m/s)

Momentum is a vector — it points in the same direction as velocity. The SI unit kg·m/s has no special name.

ELI-10: Explain It Like I'm 10

Momentum is how hard it is to stop something. A fast-moving bowling ball is harder to stop than a slow-moving one. A heavy truck is harder to stop than a bicycle at the same speed. Momentum combines both mass and speed into one number (with a direction). The more momentum something has, the more "oomph" it carries.

Momentum is especially useful for understanding recoil and explosions. When a cannon fires a cannonball forward, the cannon itself recoils backward. The forward momentum of the cannonball is exactly balanced by the backward momentum of the cannon — the total momentum of the system (cannon + ball) was zero before firing and remains zero after. This is why rockets work in the vacuum of space: they expel exhaust gases backward and gain forward momentum in exact proportion.

4.2 Impulse

Core Idea

Impulse is the product of force and the time over which it acts. Impulse equals the change in momentum.

Physics and Mathematics

J = F_avg Δt = Δp

Where:

• J = impulse (N·s, equivalent to kg·m/s)
• F_avg = average force (N)
• Δt = time interval (s)

Graphically: Impulse is the area under a force-versus-time curve.

Key insight: For a given change in momentum, a longer time interval means a smaller average force. This is why airbags, crumple zones, and padded flooring reduce injury — they increase Δt and therefore reduce F_avg.

Worked Example: Catching a Ball

Problem: A 0.15 kg baseball moving at 40 m/s is caught, and the catcher's glove brings it to rest in 0.050 s. Find the average force.

Solution: Δp = 0 - (0.15)(40) = -6.0 kg·m/s F_avg = ΔpΔt = -6.00.050 = -120 N

Answer: 120 N (about 27 lbs). If the catch took 0.005 s (stiff glove), the force would be 1,200 N — ten times higher.

ELI-10: Explain It Like I'm 10

Jump off a chair onto a hard floor and it hurts. Jump onto a soft mattress and it does not. Your momentum change is the same either way — you stop. But the mattress gives you more time to stop, so the force on your body is smaller. This is the impulse-momentum idea: spreading the stop over a longer time makes the force gentler.

4.3 Conservation of Momentum

Core Idea

In an isolated system (no net external force), the total momentum is constant.

∑p_before = ∑p_after

This is a vector conservation law — it applies separately in the x-, y-, and z-directions. If there is no net external force in a particular direction, momentum in that direction is conserved.

ELI-10: Explain It Like I'm 10

Two people on roller skates push off each other. Before the push, both are at rest — total momentum is zero. After the push, they move apart. If one is twice as heavy, they move at half the speed of the lighter person, keeping the total momentum at zero. The push does not create momentum out of nowhere — it just redistributes what was already there.

4.4 Elastic Collisions

Core Idea

In an elastic collision, both momentum and kinetic energy are conserved.

m₁ v_1i + m₂ v_2i = m₁ v_1f + m₂ v_2f  (momentum) 12m₁ v_1i² + 12m₂ v_2i² = 12m₁ v_1f² + 12m₂ v_2f²  (kinetic energy)

Special case — equal masses, 1D: The objects simply exchange velocities. If one is initially at rest, the incoming object stops and the stationary one moves away with the incoming object's speed.

Special case — 1D, target at rest: v_1f = m₁ - m₂m₁ + m₂v_1i,   v_2f = 2m₁m₁ + m₂v_1i

ELI-10: Explain It Like I'm 10

Elastic collisions are like perfectly bouncy balls. A billiard ball hitting a stationary billiard ball: the first ball stops, the second ball rolls away with the same speed. In a perfectly elastic collision, no energy is lost to heat or sound — everything bounces perfectly. Real collisions are never perfectly elastic, but hard steel balls and billiard balls come close.

Worked Example: Elastic Collision in 1D

Problem: A 2.0 kg cart moving at 3.0 m/s to the right collides elastically with a 1.0 kg cart moving at 2.0 m/s to the left. Find the final velocity of each cart.

Given: m₁ = 2.0 kg, v_1i = +3.0 m/s, m₂ = 1.0 kg, v_2i = -2.0 m/s.

Physics Principle: Both momentum and kinetic energy are conserved in an elastic collision.

Momentum: 2.0(3.0) + 1.0(-2.0) = 2.0 v_1f + 1.0 v_2f → 4.0 = 2v_1f + v_2f

Kinetic energy: 12(2.0)(3.0)² + 12(1.0)(2.0)² = 12(2.0)v_1f² + 12(1.0)v_2f² → 11.0 = v_1f² + 0.5 v_2f²

Solving the system yields: v_1f ≈ -0.33 m/s (reverses direction), v_2f ≈ +4.7 m/s (reverses and speeds up).

Does it Make Sense? The lighter cart reverses direction and gains speed; the heavier cart slows down and may reverse. This is characteristic of an elastic collision — kinetic energy is preserved, so speeds are higher after than in a comparable inelastic case.

Momentum vs. Kinetic Energy — Critical Distinction

Students often confuse these two conserved quantities. Here is the definitive comparison:

Key insight: Momentum conservation always holds (no external forces). Kinetic energy conservation is a special case. When two cars crash, momentum is conserved but kinetic energy decreases — the "lost" energy becomes heat, sound, and deformation. This is the hallmark of an inelastic collision.

Common Misconceptions

• "Momentum and kinetic energy are the same thing." They are fundamentally different quantities measured in different units. A 1 kg object at 1 m/s has momentum of 1 kg·m/s and KE of 0.5 J. They cannot be compared directly.
• "Energy is always conserved, so collisions are always elastic." Total energy IS always conserved. But mechanical kinetic energy is not conserved in inelastic collisions — it converts to thermal and other forms.
• "If momentum is conserved, the objects must have the same mass." Conservation means the vector sum is constant, not that individual momenta are equal.

4.5 Inelastic Collisions

Core Idea

In an inelastic collision, momentum is conserved but kinetic energy is NOT. Some kinetic energy is converted to thermal energy, sound, or deformation.

Perfectly inelastic collision: The objects stick together after collision. Momentum conservation gives: m₁ v_1i + m₂ v_2i = (m₁ + m₂) v_f

The kinetic energy lost is maximum in a perfectly inelastic collision (for a given initial state).

Worked Example: Car Crash

Problem: A 1500 kg car traveling at 20 m/s east collides with a 1000 kg car at rest. They stick together. Find the final velocity and the kinetic energy lost.

Momentum: 1500(20) + 0 = (2500)v_f → v_f = 12 m/s east.

Initial KE: 12(1500)(20)² = 300,000 J

Final KE: 12(2500)(12)² = 180,000 J

KE lost: 120,000 J (converted to deformation, heat, sound).

ELI-10: Explain It Like I'm 10

Inelastic collisions are like a lump of clay hitting a wall — it squishes and stops. The total "quantity of motion" (momentum) is still conserved, but some of the motion-energy becomes heat and deformation. If two cars crash and stick together, they keep moving (momentum is conserved), but they are crumpled and hot because some energy transformed.

4.6 Center of Mass

Core Idea

The center of mass is the weighted average position of all mass in a system. It moves as if all mass were concentrated there and all external forces acted there.

Physics and Mathematics

For point masses: r_cm = m₁r₁ + m₂r₂ + ⋯m₁ + m₂ + ⋯

The velocity of the center of mass relates to total momentum: v_cm = p_totalM_total

In an isolated system, v_cm is constant — the center of mass moves with constant velocity even if the objects are colliding and flying apart.

Worked Example: Center of Mass for Two Particles

Problem: A 4.0 kg mass is at x = 0 m and a 2.0 kg mass is at x = 3.0 m. Find the center of mass.

Solution: x_cm = (4.0)(0) + (2.0)(3.0)4.0 + 2.0 = 6.06.0 = 1.0 m. The center of mass is 1.0 m from the origin, closer to the heavier mass — exactly as physical intuition suggests.

Limiting Cases for Momentum and Collisions

• What if one mass is much larger than the other? In a collision with m₁ ≫ m₂ and v_2i = 0: the heavy object barely changes speed, and the light object bounces away at roughly twice the heavy object's speed. Think of a tennis ball hitting a moving truck.
• What if the collision is perfectly inelastic? Maximum kinetic energy is lost (converted to other forms). The objects stick together and move with a common final velocity.
• What if external forces are present? Momentum is NOT conserved in that direction. A ball rolling on a surface with friction loses momentum to the Earth.
• What if the system is not isolated but the collision is very brief? During the brief collision, external forces (like friction) may be negligible compared to the huge collision forces. Momentum is approximately conserved during the collision itself.

ELI-10: Explain It Like I'm 10

The center of mass is the "balance point" of an object or system. If you threw a spinning wrench through the air, the wrench would tumble crazily — but one point on it (the center of mass) would follow a smooth, predictable arc, exactly like a tossed ball. That point is the center of mass.

Topic Summary

• Momentum p = mv is a conserved vector quantity. Unlike energy, momentum has direction and is conserved separately in each dimension.
• Impulse J = FΔt = Δp. The same momentum change can be achieved with a large force over a short time or a small force over a long time — this is the engineering principle behind airbags, crumple zones, and padded surfaces.
• Momentum conservation holds in isolated systems (no net external force). It is a VECTOR law — if there is no net force in the x-direction, momentum in the x-direction is conserved even if y-momentum is not.
• Elastic collisions conserve both momentum and kinetic energy. In 1D with equal masses, the objects exchange velocities.
• Inelastic collisions conserve momentum but NOT kinetic energy. The "lost" kinetic energy converts to thermal energy, sound, and deformation. Perfectly inelastic collisions (objects stick) lose the maximum possible kinetic energy.
• Center of mass moves as a single particle under the net external force. In an isolated system, v_cm is constant regardless of internal collisions or explosions.

Essential Equations

Concept Check

1. A clay ball and a rubber ball of equal mass hit a wall at the same speed. Which imparts a greater impulse to the wall? Why?
2. Explain why a bullet fired from a rifle causes the rifle to recoil. Is momentum conserved? Is kinetic energy conserved?
3. Two identical objects collide elastically in 1D. One is initially at rest. Describe their velocities after the collision.
4. Why does an airbag reduce injury in a car crash? Use the impulse-momentum theorem.
5. In a perfectly inelastic collision, where does the "lost" kinetic energy go?
6. Can a system have zero total momentum but nonzero total kinetic energy? Give an example.
7. A neutron (mass m) collides elastically head-on with a stationary carbon nucleus (mass 12m). Approximately what fraction of the neutron's initial kinetic energy is retained after the collision?

Open Educational References

• OpenStax, College Physics, Chapter 8: Linear Momentum and Collisions
• OpenStax, University Physics, Volume 1, Chapter 9: Linear Momentum and Collisions