Momentum and impulse
The momentum of an object is its mass times its velocity: p = m v. It is a vector and has units of kg·m/s. Newton's second law can be written as net force equals the rate of change of momentum. Rearranging gives the impulse-momentum theorem: F Δt = Δp = m Δv. The product of force and the time it acts is the impulse. This explains why an airbag or a bent knee helps: extending the stopping time lowers the average force for the same change in momentum.
Conservation of momentum
If no net external force acts on a system, its total momentum stays the same. Internal forces between the objects in a collision are equal and opposite (Newton's third law), so they change each object's momentum but not the total. You must still keep track of direction: choose a positive direction and give velocities in the opposite direction a negative sign.
Types of collision
- Perfectly inelastic: the objects stick together and move with a common velocity v = (m₁v₁ + m₂v₂) / (m₁ + m₂). Maximum kinetic energy is lost for a collision that conserves momentum.
- Inelastic: the objects bounce, but some kinetic energy is lost to heat, sound and deformation.
- Elastic: kinetic energy is also conserved. In a head-on elastic collision between equal masses, with one initially at rest, the moving mass stops and the other leaves with its speed.
Momentum is conserved in all three cases; kinetic energy is conserved only in the elastic one. Explosions and recoil work the same way in reverse: a system starting at rest has zero total momentum, so the pieces move apart with momenta that cancel.
Centre of mass
The centre of mass of point masses on a line is x = (m₁x₁ + m₂x₂ + …) / (m₁ + m₂ + …). It lies closer to the heavier mass.
Worked example
A 0.20 kg cart at 3.0 m/s hits a stationary 0.30 kg cart, and they couple.
- Momentum before: 0.20 × 3.0 = 0.60 kg·m/s.
- Common speed: v = 0.60 / (0.20 + 0.30) = 1.20 m/s.
- Kinetic energy before: ½ × 0.20 × 3.0² = 0.90 J. After: ½ × 0.50 × 1.2² = 0.36 J.
- Energy lost: 0.54 J, which is the energy turned into heat and deformation.
Common mistakes
- Ignoring direction when velocities oppose.
- Assuming kinetic energy is conserved in every collision.
- Using the speed of the object after the collision in the momentum before.
- Confusing impulse (force times time) with work (force times distance).