Two carts crash. One sticks to the other, they crumple, they slow down. Surely something is lost? Something is: kinetic energy. But one quantity survives every crash untouched, whether the carts bounce cleanly apart or jam together into one mangled lump: their total momentum. Roll the carts together in the lab above and watch the green total-momentum arrow. It is the same length before and after the hit, every single time. That is the law this lesson is about.
What is momentum? The p = mv formula
Momentum measures how much motion an object carries: how hard it is to stop. It depends on both how heavy the object is and how fast it is going, combined in one short formula.
momentum: p = m × v (mass in kg, velocity in m/s, so p is in kg·m/s)
A loaded truck rolling slowly can carry more momentum than a fast bicycle, because its huge mass outweighs the bicycle’s higher speed. Momentum is a vector: it has direction. If you call rightward positive, then a cart moving left has negative momentum, and you must keep the sign when you add momenta together. That sign is why the total momentum arrow in the lab can point left, point right, or shrink to nothing when two equal carts run head-on.
A quick calculation: a 2 kg cart moving at 3 m/s has p = 2 × 3 = 6 kg·m/s. Double its speed to 6 m/s and its momentum doubles to 12 kg·m/s. Send an identical cart the other way at 3 m/s and its momentum is -6 kg·m/s, the exact opposite.
The law of conservation of momentum
Set two carts moving, let them collide, and add up the momentum before and after. In an isolated system (no outside push or pull like friction or a hand) the answer never changes:
total p before = total p after m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Here u is a velocity before the collision and v is the velocity after. This is the law of conservation of momentum, and it holds for every collision: gentle or violent, bouncy or sticky. In the lab, the “Momentum before” and “Momentum after” readouts match to the last digit, and the two momentum bars are always the same height.
Elastic collisions: momentum and kinetic energy both conserved
A perfectly elastic collision is the bounciest kind: the objects rebound with no loss of kinetic energy at all. In the lab this is the elasticity slider at e = 1. Two things are conserved here, not one: total momentum and total kinetic energy.
kinetic energy: KE = ½ × m × v²
Real-world collisions are almost never perfectly elastic (steel ball bearings and billiard balls come close), but the elastic case is the clean benchmark. The most famous example is a Newton’s cradle: one ball swings in, one ball swings out, at the same speed.
That result surprises people who expect the heavier-looking, still-moving cart A to plough through. With equal masses and a perfect bounce, cart A hands over all of its momentum and stops dead. Following the preset, the total momentum stayed 8 kg·m/s for the 2 kg cart at 4 m/s, and the total kinetic energy stayed 16 J: both conserved.
Inelastic and perfectly inelastic collisions: energy is lost
In an inelastic collision the objects do not bounce back perfectly. Some kinetic energy is converted into heat, sound, and permanent deformation (the crumple of a car bumper). Momentum is still conserved, but kinetic energy is not. On the slider this is any e less than 1.
The extreme case is a perfectly inelastic collision, e = 0, where the objects stick together and move off as one. This loses the largest possible fraction of kinetic energy while still conserving momentum. Because the two bodies share a single final velocity, you can find it directly:
perfectly inelastic (stick): v = (m₁u₁ + m₂u₂) / (m₁ + m₂)
Worked example: a 2 kg cart at 4 m/s hits a stationary 2 kg cart with e = 0. They move off together at v = (2×4 + 2×0) / (2 + 2) = 2 m/s. Total momentum is 8 kg·m/s before and after, so momentum is conserved. But kinetic energy falls from ½ × 2 × 4² = 16 J to ½ × 4 × 2² = 8 J: half the energy is gone, turned into sound and deformation. Set exactly this up in the lab and the KE-after bar drops to half height while the momentum bars stay level.
Elastic vs inelastic collisions: the one real difference
It is tempting to think elastic and inelastic collisions differ in whether momentum is conserved. They do not. Momentum is conserved in both. The single real difference is kinetic energy.
| Collision type | e | Momentum | Kinetic energy |
|---|---|---|---|
| Perfectly elastic | 1 | Conserved | Conserved |
| Inelastic | between 0 and 1 | Conserved | Some lost |
| Perfectly inelastic (stick) | 0 | Conserved | Most lost |
The coefficient of restitution e ties them together. It is the ratio of the separation speed after the collision to the approach speed before, a number from 0 (stick) to 1 (perfect bounce). Slide e from 1 down to 0 in the lab and watch the momentum bars refuse to change while the KE-after bar shrinks steadily. That contrast is the whole idea in one picture.
Impulse: how a force changes momentum
To change an object’s momentum you have to push on it for some time. That combined effect, force acting over time, is called impulse, and it equals the change in momentum. This is the impulse-momentum theorem:
impulse: J = F × t = Δp = m × v - m × u
Worked example: hit a 0.5 kg ball with an average force of 20 N for 0.1 s. The impulse is J = 20 × 0.1 = 2 N·s, which equals the ball’s change in momentum, so its momentum rises by 2 kg·m/s. Starting from rest, that is a final speed of 4 m/s.
Impulse is also why momentum is conserved in a collision. During contact the two carts push on each other with equal and opposite forces (Newton’s third law) for the same short time, so they receive equal and opposite impulses. One cart’s momentum gain exactly matches the other’s loss, and the total does not budge. The lab shows this directly: the “Impulse on B” and “Impulse on A” readouts are always equal in size and opposite in sign.
Common momentum mistakes
- “Momentum is lost when things crash or stick.” No. Total momentum is conserved in every collision in an isolated system. The stuck-together, slowed-down carts in the e = 0 run have the exact same total momentum as before.
- “Elastic and inelastic collisions differ in whether momentum is conserved.” Both conserve momentum. The real difference is whether kinetic energy is conserved (only at e = 1).
- “An energy-losing collision breaks conservation.” Losing kinetic energy and conserving momentum happen at the same time. The verdict card holds the two ideas side by side on purpose.
- “Momentum is just speed, or just energy.” Momentum p = m×v is a vector that can be negative; kinetic energy KE = ½m×v² is a scalar that is never negative. A cart moving left has negative momentum but positive energy.
- “The heavier or still-moving object always pushes through.” In the equal-mass elastic swap, the incoming cart stops dead and the target leaves with all the speed. Direction and mass, not intuition, decide the outcome.
Keep exploring
The equal-and-opposite forces behind conservation are Newton’s third law, and the force-over-time idea in impulse is the same F that drives the F = ma calculator. See where kinetic energy actually goes when it is “lost” to rubbing surfaces in friction, and turn the carts’ constant-velocity travel before and after the hit into slopes on a motion graph.