Choosing Between Momentum and Energy
- State the distinct conditions under which momentum and mechanical energy are conserved
- Identify which conservation law a given problem allows and which it forbids
- Compute the mechanical energy lost in a perfectly inelastic collision
Two laws, two different conditions
Momentum is conserved when the net external force is zero. Mechanical energy is conserved when no non-conservative force does work. These are genuinely different requirements, and this is the source of most confusion in the unit. Two carts colliding and sticking together satisfy the first but not the second: no external force acts, so momentum survives, while the deformation of the bumpers turns kinetic energy into heat and sound.
The collision spectrum
Every collision conserves momentum. They differ in what happens to kinetic energy. Elastic: kinetic energy is fully conserved — a good approximation for hard steel balls, billiard balls and gas molecules. Inelastic: some kinetic energy becomes thermal, sound and deformation. Perfectly inelastic: the objects stick together and move as one, and this loses the maximum kinetic energy consistent with conserving momentum. Note that "maximum loss" does not mean "all of it" — the combined object still moves, so some kinetic energy always survives unless the total momentum was zero.
A 4.0 kg cart moving at 6.0 m/s collides with and sticks to a stationary 2.0 kg cart. Find the final speed and the kinetic energy lost.
- 1.Momentum before: p = (4.0)(6.0) = 24 kg·m/s. Momentum is conserved in every collision.
- 2.After, the combined 6.0 kg moves at v = 24 ÷ 6.0 = 4.0 m/s.
- 3.Kinetic energy before: ½(4.0)(6.0)² = 72 J.
- 4.Kinetic energy after: ½(6.0)(4.0)² = 48 J.
- 5.Loss: 72 − 48 = 24 J, which became thermal energy and sound.
The decision procedure
Read the problem and ask two questions. First: is anything external pushing on the system during the interval? If not, write the momentum equation — it is always available in a collision. Second: does the problem say elastic, or does anything deform, stick, scrape or make noise? Only if the answer is a clean "elastic" may you also write the kinetic-energy equation. Two unknowns need two equations, which is exactly why elastic collision problems give you both and inelastic ones give you the sticking condition instead.
A useful bridge: K = p²/2m
Substituting p = mv into K = ½mv² gives K = p²/2m. This relation settles a family of exam questions instantly. Two objects with the same momentum: the lighter one has more kinetic energy. Two objects with the same kinetic energy: the heavier one has more momentum. Neither result is obvious from the separate formulas, and both appear regularly in multiple choice.
"Energy is conserved" is always true; "mechanical energy is conserved" often is not. In an inelastic collision the missing kinetic energy has not disappeared — it is thermal energy in the deformed metal and sound in the air. Say where it went, not that it was lost.
A bullet embeds itself in a wooden block on a frictionless surface. Which is conserved?
Two objects have equal momentum but object A has twice the mass of object B. Which has more kinetic energy?
In which collision is the most kinetic energy lost, for the same initial conditions?
When a free-response has a collision followed by a swing, slide or rise, it almost always wants momentum for the collision and energy for what follows. Using energy through the collision itself is the single most common way students lose the whole question.
Answer the 3 checkpoints as you read.
Sign in to save your progress