Collisions include elastic collisions, in which energy is conserved, and inelastic collisions, in which energy is lost. Master the difference between them and how to solve them.
Basic Knowledge
Elastic collision (perfectly elastic collision): Both conservation of momentum and conservation of kinetic energy hold.
Solving the simultaneous equations (masses m₁, m₂; velocities before collision v1, v2):v1 = (m₁−m₂)v1 + 2m₂v2 / (m₁+m₂)v2 = 2m₁v1 + (m₂−m₁)v2 / (m₁+m₂)
Special case (m₁ = m₂, v2 = 0): the velocities are exchanged (v1 = 0, v2 = v1)
Inelastic collision: Only conservation of momentum holds. Some kinetic energy is lost (as heat, sound, or deformation).
Perfectly inelastic collision (objects stick together): v' = (m₁v1 + m₂v2)/(m₁+m₂)
Energy loss: ΔE = ½m₁v1² + ½m₂v2² − (½m₁v1² + ½m₂v2²) ≥ 0
Elastic collision: conservation of momentum + conservation of energy (two simultaneous equations)
Perfectly inelastic collision: v' = (m₁v1 + m₂v2)/(m₁+m₂)
Elastic collision of equal masses: velocities are exchanged
Energy loss: difference in KE before and after collision (≥ 0)
Elastic condition: ½m₁v1² + ½m₂v2² = ½m₁v1² + ½m₂v2²
Further Study (Background and Significance)
When solving the simultaneous equations for an elastic collision, the energy-conservation equation contains squared velocities, making direct solution complicated. Instead, using the equivalent condition of reversal of relative velocity makes the calculation easier (this corresponds to coefficient of restitution e = 1).
Collisions at the atomic and molecular levels are close to ideal elastic collisions and form the basis of the kinetic theory of gases. In contrast, everyday collisions (cars, balls, and people) are inelastic to varying degrees.
Energy loss is greatest in a perfectly inelastic collision. The lost energy can also be written as ΔE = μΔv²/2 (μ is the reduced mass m₁m₂/(m₁+m₂), and Δv is the relative velocity).
Newton's cradle, an apparatus consisting of a row of metal balls, visually demonstrates the exchange of velocities in elastic collisions between equal masses.
- Elastic: both momentum and energy are conserved
- Inelastic: only momentum is conserved (KE decreases)
- Perfectly inelastic (sticking together): maximum KE loss
- Elastic collision between equal masses: velocities are exchanged
- Equivalent condition for elastic collision: relative velocity reverses
- ΔE ≥ 0 (energy does not increase)
- Two equations (two conservation laws) → two unknowns
Points to Note (Common Confusions)
① It is common to remember “momentum is conserved in collisions,” but energy is also conserved only in elastic collisions. ② The proper method for elastic-collision calculations is to solve the two conservation equations simultaneously. ③ A perfectly inelastic collision loses the most energy (the final KE is smallest). ④ Energy loss is always ≥ 0. KE cannot increase after a collision without work from outside.
Practice
- A ball of mass m elastically collides with an identical stationary ball. Find the velocities of both balls after the collision.
- An object of mass 1.0 kg moving at 4.0 m/s undergoes a perfectly inelastic collision with a stationary object of mass 3.0 kg. Find the velocity after the collision and the kinetic energy lost.
- What conservation law holds for both elastic and inelastic collisions?