At a glance
Two carts collide head-on on a frictionless track, and the sim keeps a ledger of momentum and kinetic energy from before the impact to after it. You set the masses, the speeds and the bounciness, and you can watch the same crash from a moving frame.
What you'll learn
- How to count momentum and kinetic energy for a collision, and how each total compares before and after.
- How the restitution e sets the rebound, from sticking (e = 0) to perfectly elastic (e = 1).
- How the kinetic energy lost depends on the masses, the approach speed and e.
- How the same collision looks from a moving frame, and which numbers depend on the frame.
Before the impact: each cart has a mass m and a velocity v (the arrow). They meet at the dashed marker.
What's on screen
- Cart m₁ (cyan), labelled with its mass and velocity. The arrow shows the velocity.
- Cart m₂ (amber), labelled the same way.
- Merged cart (violet): the two carts stuck together, or the parent body of an explosion.
- Impact marker (dashed line): where the carts meet. The label at top left names the frame: Lab frame, or the moving frame's vb.
- Momentum & energy: m, v, p and KE for each cart, before and after, with totals.
- Frame invariants and Energy ledger: Σp and ΔKE, and ΣKE before and after.
See itRun m₁ = 2 kg at +3 m/s into m₂ = 1 kg at −1 m/s, with e = 0.5. Watch for: the carts meeting at the marker and leaving with new velocities, shown on screen and in the card below.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Mode | Inelastic: you choose e. Elastic: e is locked at 1. Explosion: one body at rest splits in two. | 3 modes opens in Inelastic | |
| m₁, m₂ | The two masses. | 0.5–5 kg open at 1.0 kg each | |
| v₁, v₂ | The two velocities before the impact. A negative value moves left. Greyed out in Explosion. | −5 to +5 m/s open at +2.0 and 0.0 | |
| Frame velocity vb | The velocity of the camera. Every v, p and KE is re-expressed in that frame. | −5 to +5 m/s opens at 0 (lab frame) | |
| Restitution e | How bouncy the impact is: 0 sticks, 1 is elastic. Active in Inelastic mode only. | 0–1 opens at 0 | |
| Energy released Q | The stored energy an explosion turns into motion. Active in Explosion mode only. | 0–10 J opens at 4.0 J | |
| Momentum & energy | Table of m, v, p and KE for each cart before and after, with the totals. | panel | |
| Frame invariants | Σp before and after, ΔKE, and the frame's vb and vcm. | panel | |
| Energy ledger | ΣKE before and after, and the difference. | panel | |
| ▶ Play / ⏸ Pause | Runs the approach, the impact and the separation once, or freezes the run. | top bar | |
| Speed | Plays the run faster or slower on screen. The numbers do not change. | 0.25×–4× opens at 1× | |
| ↻ Reset | Puts every control back to its opening value and pauses at the start of the run. | top bar |
Fixed: a frictionless track, with the carts always starting on opposite sides of the impact marker. The faster cart starts on the left, whichever label it has.
Momentum is conserved in every collision
Momentum has a sign, because velocity does. Kinetic energy is never negative. The sim totals both over the two carts.
The two quantities in every row of the table.
During contact the carts push on each other with equal and opposite forces for the same time. Their impulses cancel, so the total momentum cannot change, whether the carts stick, bounce or fly apart.
u is a velocity before the impact (the v₁ and v₂ sliders). v′ is a velocity after.
- Σp can be positive, negative or zero, because the carts' momenta can cancel.
- A stuck pair (e = 0) moves with v′ = Σp / (m₁ + m₂). If Σp = 0, the pair stops. That is momentum that was zero all along, not momentum destroyed.
See itStick m₁ = 2 kg at +3 m/s onto m₂ = 1 kg at rest. Watch for: Σp reading 6.000 kg·m/s before and after, and the stuck pair moving at +2.0 m/s.
Restitution sets the bounce
Two unknown final velocities need two equations. Momentum conservation gives one. The restitution e gives the other: the speed of separation is e times the speed of approach.
urel = u₁ − u₂ is the approach speed before the impact. v′rel is the relative velocity after, so it points the other way.
- e = 0: the carts do not separate, so they stick.
- e = 1: perfectly elastic. Separation speed equals approach speed, and kinetic energy is conserved too. Elastic mode is this case.
- 0 < e < 1: they separate more slowly than they approached.
The same head-on start (1 kg each, ±2 m/s) at three settings of e.
See itRun two 1 kg carts at +2 and −2 m/s with e = 0.5. Watch for: the carts separating at 2.0 m/s relative to each other, half the 4.0 m/s they approached at.
See itElastic mode: m₁ = 1 kg at +2 m/s hits an identical m₂ at rest. Watch for: m₁ stopping dead and m₂ leaving at +2.0 m/s. Equal masses swap velocities.
How much kinetic energy is lost
Total kinetic energy splits into the motion of the centre of mass, ½Mvcm², and the motion relative to it, ½μurel², where μ = m₁m₂/(m₁ + m₂) is the reduced mass. Σp is conserved, so vcm cannot change. Only the relative part can: it starts at ½μurel² and ends at ½μe²urel².
For equal masses, μ = m/2.
m₁
m₂
u₁ (v₁)
u₂ (v₂)
e
These sliders move the sim too.
- The loss grows as urel²: double the approach speed and it quadruples.
- It scales with 1 − e², not 1 − e. At e = 0.5 the loss is still 75% of the sticking loss.
- The sim reports how much kinetic energy is lost, not where it goes (heat, sound, permanent deformation).
See itRun the same head-on pair (1 kg each, ±2 m/s) at e = 0, 0.5 and 1. Watch for: three runs in turn, and the card listing ΔKE for each: −4, −3 and 0 J.
What a moving frame changes
Watched from a frame moving at vb, every velocity becomes v′ = v − vb. That is what the Frame velocity slider does.
Here Σp′ and ΔKE′ are measured in the moving frame; vb is its velocity in the lab.
- Σp shifts by (m₁ + m₂)vb, by the same amount before and after. So it is still conserved in every frame.
- ΔKE does not change, because it depends only on urel and a boost leaves urel alone.
- At vb = vcm = Σp / (m₁ + m₂), the total momentum is zero. A stuck pair then stops.
See itStick two 1 kg carts at +2 and −2 m/s, watched from vb = +1.5 m/s. Watch for: Σp reading −3.000 kg·m/s before and after, while ΔKE still reads −4.000 J.
See itStick m₁ = 2 kg at +3 m/s onto m₂ = 1 kg at rest, with two cameras. Watch for: the lab view above carrying the stuck pair on at +2.0 m/s, and the view below, moving at vcm = +2.0 m/s, showing it at rest.
An explosion is a collision run backwards
One body of mass M = m₁ + m₂ starts at rest and splits in two. Stored energy Q turns into kinetic energy, and Σp stays 0, so the fragments move off with equal and opposite momenta p*.
μ = m₁m₂/(m₁ + m₂), the same reduced mass as in a collision.
- Since KE = p²/2m, the lighter fragment carries the larger share of Q: KEA / KEB = mB / mA.
- The sim treats Q as released from a store (a spring, a chemical charge). It does not model what the store is.
See itExplode a body into m₁ = 1 kg and m₂ = 3 kg with Q = 4 J. Watch for: the 1 kg fragment leaving at +2.4 m/s and the 3 kg fragment at −0.8 m/s, with Σp still 0.
Where it matters
- Ballistic pendulum. A bullet embeds in a hanging block. Momentum conservation gives the speed just after the impact, and energy conservation after it gives the swing height. A 10 g bullet at 100 m/s into a 1.0 kg block leaves the pair at 0.99 m/s, rising 5.0 cm. Only about 1% of the bullet's kinetic energy is left to swing.
- Crumple zones. A car body is built to deform. The kinetic energy goes into bending metal, and the longer stopping time lowers the average force on the passengers (F = Δp/Δt).
- Neutron moderation. In a reactor, neutrons slow down by hitting light nuclei. Treat the collision as elastic and head-on, with the nucleus at rest: the neutron can hand over at most 4m₁m₂/(m₁ + m₂)² of its energy. That is nearly all of it for a proton (almost the same mass as a neutron), and under 30% for carbon-12.
Go deeper: the elastic closed forms
The e = 1 solution, which Elastic mode uses.
- Equal masses swap velocities: v₁′ = u₂ and v₂′ = u₁.
- A heavy cart hitting a light one at rest gives it nearly 2u₁. With m₁ = 5 kg, m₂ = 0.5 kg and u₁ = 3 m/s the sim shows v₂′ = 5.45 m/s.
- A light cart hitting a heavy one at rest bounces back with nearly −u₁.
Summary
In one line. Σp is the same before and after every collision, whatever e is. Kinetic energy is lost by an amount ½μ(1 − e²)urel², which every frame agrees on.
Key points
- Equal and opposite contact forces keep Σp fixed. e then fixes the separation speed.
- e = 0 sticks, e = 1 is elastic. Only elastic collisions conserve kinetic energy.
- The kinetic energy lost depends on μ, urel and e, and not on the frame.
- Σp depends on the frame. It is zero in the centre-of-mass frame.
Common mix-ups
- "If kinetic energy vanishes when carts stick, momentum must vanish too." Internal forces cancel in Σp, and kinetic energy has somewhere to go (heat, deformation).
- "At e = 0.5 half of the sticking loss is left." The loss scales with 1 − e², so 75% of it is lost.
- "Momentum is conserved only in the lab frame." It is conserved in every Galilean frame. Its value changes with the frame, and its constancy does not.
Things to try on your own
- Choose masses and speeds that give Σp = 0, set e = 0, and read the Momentum & energy table after the impact.
- Set m₁ = 5 kg, m₂ = 0.5 kg, v₂ = 0 in Elastic mode, and compare v₂′ with 2v₁.
- With e = 0 and unequal masses, drag the Frame velocity slider until Σp after reads 0, and compare vb with vcm.