Momentum and collisions
p = m v
Momentum and the two collisions of the chapter, on an oriented line where the sign is part of the speed. The perfectly inelastic collision sticks the two bodies together: momentum imposes their common speed, and the kinetic energy lost to deformation and heat is quantified, in joules and as a percentage. The elastic collision conserves both quantities at once and returns the two final speeds, with the limiting cases that teach: equal masses, the speeds swap, the pétanque carreau; a massive wall, the ball bounces back at opposite speed.
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)
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 swapped, exactly. It is the pétanque carreau, and the reason a billiard ball stops dead when hitting another head-on.
Momentum is conserved in every collision, elastic or not: no external force acts during the instant of contact. Kinetic energy is only conserved in the elastic case; elsewhere it leaves as deformation, heat and sound, and that loss is a result, not an error.
Everything happens on an oriented line: a negative speed goes to the left. That convention handles head-on collisions with no special case, the sign being part of the speed.
The formulas hold for a ONE-dimensional, head-on, centred collision. A two-dimensional collision (off-centre billiard balls) conserves the momentum vector component by component, and requires the angles.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasp = m v
p = m v
conservation: m₁v₁ + m₂v₂ = m₁v₁’ + m₂v₂’
inelastic: v’ = (m₁v₁ + m₂v₂)/(m₁ + m₂)
elastic: v₁’ = ((m₁−m₂)v₁ + 2m₂v₂)/(m₁+m₂)
Momentum is conserved in every collision, because no external force acts during the instant of contact: it is the most robust law of the chapter. Kinetic energy is only conserved in the elastic case; its loss elsewhere is not an error but a result, that of deformation, heat and sound.
- p
- · momentum, m × v, in kg·m/s. A SIGNED quantity on the oriented line.
- Elastic collision
- · the bodies bounce without loss: p and kinetic energy are both conserved.
- Perfectly inelastic collision
- · the bodies leave stuck together, at a common speed: the energy loss is then the largest possible.
- Relative speed
- · in the one-dimensional elastic case it reverses exactly: 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 are needed. Real collisions sit between elastic and perfectly inelastic, described by an intermediate coefficient of restitution; the two modes of the tool are its exact bounds. Speeds stay far below light.
Reading the resultThe limiting cases say it all: with equal masses the speeds swap, which is why a billiard ball stops dead on a head-on hit; against an immense mass, the light body bounces back at opposite speed, and that is the atom against the vessel wall that makes gas pressure.
The limiting cases say it all: with equal masses the speeds swap, which is why a billiard ball stops dead on a head-on hit; against an immense mass, the light body bounces back at opposite speed, and that is the atom against the vessel wall that makes gas pressure. In the symmetric inelastic case everything stops and all the energy is lost, while momentum, zero before, stays zero after: the two conservations do not tell the same story.