Collision Lab
Two carts collide on a track; momentum, energy and the centre of mass update live.
About this simulation
Set each cart's mass and velocity and the restitution e, from perfectly elastic to perfectly inelastic. Momentum is conserved every time; kinetic energy only when e = 1. The centre-of-mass marker glides through the collision at constant velocity — the anchor students most often miss.
How to use it
- Set Mass 1, Mass 2, Velocity 1 and Velocity 2, then watch the carts meet on the track; a negative velocity means moving left.
- Move the Restitution slider from 1 down to 0 and watch 'KE lost' at the top after each impact, with the total momentum before → after beside it.
- Switch Arrows from Velocity to Momentum and see the two momentum arrows change while their total stays fixed during the collision.
- Follow the dashed centre-of-mass marker and read its speed in the note under the track.
- Try the presets 'Equal masses swap', 'They stick (e = 0)' and 'Heavy into light'.
What to notice
- The total momentum readout stays the same when the carts hit each other, for every value of the restitution e.
- Kinetic energy is conserved only when e = 1; for smaller e some is lost in the impact, and at e = 0 the carts move off together.
- The centre of mass keeps the same velocity through the carts' collision, because the forces they exert on each other are internal.
- When a cart bounces off an end wall the total momentum does change, because the wall exerts an external force.
The formulas
- m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Conservation of momentum: u is the velocity before the collision and v after, for carts of mass m₁ and m₂.
- e = (v₂ − v₁) / (u₁ − u₂)
- Coefficient of restitution: speed of separation divided by speed of approach.
- v_cm = (m₁v₁ + m₂v₂) / (m₁ + m₂)
- Velocity of the centre of mass of the two carts.
Questions
What is the difference between elastic and inelastic collisions?
In both kinds of collision the total momentum is conserved. In an elastic collision the total kinetic energy is also conserved, so the bodies bounce apart with the same relative speed. In an inelastic collision some kinetic energy turns into heat, sound or deformation. A perfectly inelastic collision is one where the bodies stick together. In the simulation, compare e = 1 with e = 0.
What is the coefficient of restitution?
The coefficient of restitution e is the ratio of the relative speed of separation after a collision to the relative speed of approach before it. It is 1 for a perfectly elastic collision, 0 for a perfectly inelastic one, and between 0 and 1 for most real collisions. In the simulation, the Restitution slider sets e and the line under the p and KE figures says whether kinetic energy is lost, and the KE figure shows how much it drops at the impact.
What happens when two equal masses collide elastically?
In a one-dimensional elastic collision between equal masses, the two bodies exchange velocities. A moving ball that hits an identical ball at rest stops dead, and the other moves off with the first ball's speed. This follows from applying conservation of both momentum and kinetic energy. In the simulation, the 'Equal masses swap' preset shows the carts trading velocities.
Is kinetic energy conserved in a perfectly inelastic collision?
No. Momentum is conserved, but kinetic energy is not. When the bodies stick together, they move with the common velocity (m₁u₁ + m₂u₂)/(m₁ + m₂), and the kinetic energy of the centre of mass motion is all that remains. The rest is lost to deformation and heat. This is the largest possible loss for the given momentum. Try e = 0 in the simulation and watch the KE figure above the track before and after the impact.
Why does the centre of mass velocity not change in a collision?
During a collision the two bodies push on each other with equal and opposite forces. These are internal forces, so they cancel when you add them for the whole system. With no external force, total momentum is constant, and since total momentum equals total mass times the centre of mass velocity, that velocity cannot change. In the simulation, watch the centre-of-mass marker glide straight through the impact.