Collisions

Two pucks on a frictionless surface: head-on or glancing, elastic through to perfectly inelastic.

Rigid body mechanics — Forces, momentum and energy acting on bodies, and what happens when they collide.

Open it in The Lab The equations behind it

Experiments already set up on it

Elastic head-on collision
Equal masses, e = 1: the moving puck stops dead and the other goes off at u₁.
- Momentum and kinetic energy are both conserved when e = 1
- Equal masses exchange velocities in a head-on elastic collision
How much energy a collision keeps
Sweep e from 0 to 1 and watch momentum hold while energy does not.
- Momentum is conserved whatever the restitution
- Energy kept is e² of the energy in the approach
Perfectly inelastic
e = 0: the pucks stick and move off together at the centre-of-mass speed.
- A common final velocity of (m₁u₁ + m₂u₂)/(m₁ + m₂)
- The largest possible energy loss still conserves momentum
Glancing collision
Equal masses, elastic, off-centre: the two leave at 90° to each other.
- Momentum is a vector: it is conserved in both directions
- Equal masses in an elastic glancing collision separate at 90°
- The impulse acts along the line joining the centres

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)
Position of puck 1x₁ (m)
Position of puck 2x₂ (m)
y of puck 1y₁ (m)
y of puck 2y₂ (m)
Gap between the pucksΔ (m)
Speed of puck 1v₁ (m/s)
Speed of puck 2v₂ (m/s)
Velocity of puck 1 along xv₁ₓ (m/s)
Velocity of puck 2 along xv₂ₓ (m/s)
Velocity of puck 1 along yv₁ᵧ (m/s)
Velocity of puck 2 along yv₂ᵧ (m/s)
Direction of puck 1θ₁ (°)
Direction of puck 2θ₂ (°)
Total momentum along xp (kg m/s)
Total momentum along yp_y (kg m/s)
Momentum of puck 1p₁ (kg m/s)
Momentum of puck 2p₂ (kg m/s)
Velocity of the centre of massv_{cm} (m/s)
Kinetic energy of puck 1E₁ (J)
Kinetic energy of puck 2E₂ (J)
Total kinetic energyEₖ (J)
Kinetic energy lostΔEₖ (J)
Fraction of kinetic energy leftE/E₀
Collisions so farn

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Mass of puck 1m₁ (kg)
Initial velocity of puck 1u₁ (m/s)
Radius of puck 1r₁ (m)
Mass of puck 2m₂ (kg)
Initial velocity of puck 2u₂ (m/s)
Radius of puck 2r₂ (m)
Coefficient of restitutione
Impact parameterb (m)
Starting separationd (m)

The equations behind it

What collisions is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Forces and momentum

$F_{net} = ma$
Newton’s second law
$p = m v$
Momentum
$F_{net} = \frac{Δ p}{Δ t}$
Force as a rate of change of momentum
$J = F Δ t = Δ p$
Impulse
$F_{f} \leq μ_{s} N$
Static friction — At most this, until it slips.
$F_{f} = μ_{d} N$
Dynamic friction

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency

Also in Rigid body mechanics

Experiments already set up on it

Elastic head-on collision

Equal masses, e = 1: the moving puck stops dead and the other goes off at u₁.

Momentum and kinetic energy are both conserved when e = 1
Equal masses exchange velocities in a head-on elastic collision

How much energy a collision keeps

Sweep e from 0 to 1 and watch momentum hold while energy does not.

Momentum is conserved whatever the restitution
Energy kept is e² of the energy in the approach

Perfectly inelastic

e = 0: the pucks stick and move off together at the centre-of-mass speed.

A common final velocity of (m₁u₁ + m₂u₂)/(m₁ + m₂)
The largest possible energy loss still conserves momentum

Glancing collision

Equal masses, elastic, off-centre: the two leave at 90° to each other.

Momentum is a vector: it is conserved in both directions
Equal masses in an elastic glancing collision separate at 90°
The impulse acts along the line joining the centres

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)

Position of puck 1x₁ (m)

Position of puck 2x₂ (m)

y of puck 1y₁ (m)

y of puck 2y₂ (m)

Gap between the pucksΔ (m)

Speed of puck 1v₁ (m/s)

Speed of puck 2v₂ (m/s)

Velocity of puck 1 along xv₁ₓ (m/s)

Velocity of puck 2 along xv₂ₓ (m/s)

Velocity of puck 1 along yv₁ᵧ (m/s)

Velocity of puck 2 along yv₂ᵧ (m/s)

Direction of puck 1θ₁ (°)

Direction of puck 2θ₂ (°)

Total momentum along xp (kg m/s)

Total momentum along yp_y (kg m/s)

Momentum of puck 1p₁ (kg m/s)

Momentum of puck 2p₂ (kg m/s)

Velocity of the centre of massv_{cm} (m/s)

Kinetic energy of puck 1E₁ (J)

Kinetic energy of puck 2E₂ (J)

Total kinetic energyEₖ (J)

Kinetic energy lostΔEₖ (J)

Fraction of kinetic energy leftE/E₀

Collisions so farn

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Mass of puck 1m₁ (kg)

Initial velocity of puck 1u₁ (m/s)

Radius of puck 1r₁ (m)

Mass of puck 2m₂ (kg)

Initial velocity of puck 2u₂ (m/s)

Radius of puck 2r₂ (m)

Coefficient of restitutione

Impact parameterb (m)

Starting separationd (m)

The equations behind it

What collisions is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Forces and momentum

$F_{net} = ma$
Newton’s second law
$p = m v$
Momentum
$F_{net} = \frac{Δ p}{Δ t}$
Force as a rate of change of momentum
$J = F Δ t = Δ p$
Impulse
$F_{f} \leq μ_{s} N$
Static friction — At most this, until it slips.
$F_{f} = μ_{d} N$
Dynamic friction

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency

Collisions

Experiments already set up on it

Elastic head-on collision

How much energy a collision keeps

Perfectly inelastic

Glancing collision

What you can measure

What you can change

The equations behind it

Forces and momentum

Work, energy and power

Also in Rigid body mechanics

Inclined plane & friction

Rotational dynamics

Moments & equilibrium

Circular motion

Collisions

Experiments already set up on it

Elastic head-on collision

How much energy a collision keeps

Perfectly inelastic

Glancing collision

What you can measure

What you can change

The equations behind it

Forces and momentum

Work, energy and power

Also in Rigid body mechanics

Inclined plane & friction

Rotational dynamics

Moments & equilibrium

Circular motion