The formula
- Linear momentum
p = mv- Conservation of momentum for two objects
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'- Perfectly inelastic collision, where the objects stick together
m₁v₁ + m₂v₂ = (m₁ + m₂)v'
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
p | Linear momentum | kg·m/s |
m | Mass of the object | kg |
v | Velocity, with sign set by your chosen positive direction | m/s |
m₁, m₂ | Masses of the two objects | kg |
v₁, v₂ | Velocities before the interaction | m/s |
v₁', v₂', v' | Velocities after the interaction | m/s |
When it applies
- The system is closed, with no net external force acting during the interaction itself.
- Momentum is conserved in every collision, elastic or not, unlike kinetic energy, which survives only an elastic one.
- Momentum is a vector, so in two dimensions apply conservation separately along each axis.
Worked example
Problem. A 0.50 kg cart moving at 4.0 m/s strikes a stationary 1.5 kg cart and the two couple together. How fast do they move afterward, and how much kinetic energy is lost?
- Momentum before: p = (0.50 kg)(4.0 m/s) + (1.5 kg)(0 m/s) = 2.0 kg·m/s.
- After the collision the carts move as one, with a combined mass of 0.50 + 1.5 = 2.0 kg.
- Apply conservation: 2.0 kg·m/s = (2.0 kg)v', so v' = 1.0 m/s in the original direction of travel.
- Compare the kinetic energies: before, ½(0.50)(4.0)² = 4.0 J; after, ½(2.0)(1.0)² = 1.0 J.
- So 3.0 J went into deformation, heat and sound, even though momentum was fully conserved.
Answer. The coupled carts move off at 1.0 m/s, and 3.0 J of the original 4.0 J of kinetic energy is lost.
Common mistakes
- Ignoring direction. A cart moving left carries negative momentum if right is positive, and the signs matter more than the magnitudes.
- Assuming kinetic energy is conserved as well. In this collision three quarters of it disappears.
- Applying conservation over an interval where an external force acts, such as friction over a long stretch or a wall pushing back.
- Adding momenta from different directions as if they were scalars. Break them into components first.
Related formulas
- Impulse formula:
J = F_avg·Δt - Work-energy theorem:
W_net = ΔKE = ½mv² - ½mv₀² - Newton's second law:
ΣF = ma