Momentum and collision calculator
p = mv, elastic and perfectly inelastic collisions in one dimension, with conservations checked and the lost energy quantified.
Result
0 m/s
- Speed 2 after the collision
- 5 m/s
- Calculation
- v₁’ = ((m₁−m₂)v₁ + 2m₂v₂)/(m₁+m₂); v₂’ = ((m₂−m₁)v₂ + 2m₁v₁)/(m₁+m₂)
- Total momentum
- 5 kg·m/s
- Kinetic energy before
- 12.5 J
- Kinetic energy after
- 12.5 J
Notes (4)
All arrows share the same speed scale. A body’s size only shows which is heavier; its mass is written underneath.
After the collision, body 1 leaves at 0 m/s and body 2 at 5 m/s, with no energy loss at all.
Equal masses: the speeds have swapped.
Momentum is conserved in every collision; kinetic energy only in the elastic case.
Calculation, assumptions and sources
OverviewMomentum p = mv and one-dimensional collisions, on an oriented line where velocity is signed.
Momentum p = mv and one-dimensional collisions, on an oriented line where velocity is signed. For the perfectly inelastic collision, the common speed of the two bodies and the kinetic energy lost, in joules and as a percentage; for the elastic collision, the two final speeds.
Method & formulasp = m v
Momentum is conserved in every collision, since no external force acts during the contact. Kinetic energy is only conserved in the elastic case; elsewhere part of it goes into deformation, heat and sound.
- p
- momentum m × v, in kg·m/s, signed on the oriented line.
- Elastic collision
- a collision that conserves p and kinetic energy.
- Perfectly inelastic collision
- the bodies leave stuck together at a common speed; the kinetic energy loss is maximal.
- Relative speed
- in a one-dimensional elastic collision, v₁’ − v₂’ = −(v₁ − v₂).
Validity domainThe formulas hold for a one-dimensional, head-on, centred collision between two bodies isolated during the contact.
The formulas hold for a one-dimensional, head-on, centred collision between two bodies isolated during the contact. An off-centre collision is two-dimensional: the momentum vector is conserved component by component and angles come in. Real collisions, with an intermediate coefficient of restitution, lie between the two modes. Speeds are far below the speed of light.
Reading the resultWith equal masses, an elastic collision swaps the speeds; a billiard ball hit head-on stops dead.
With equal masses, an elastic collision swaps the speeds; a billiard ball hit head-on stops dead. Against a much larger mass, the light body bounces back at opposite speed. In the symmetric inelastic case, momentum is zero before and after, and all the kinetic energy is lost.