Physics HL · Chapter 4: Linear Momentum
4.3 Collision Models: Elastic, Inelastic, and Total Inelastic
Compare collision classes using momentum conservation plus kinetic-energy accounting.
Estimated time: 40 minutes
Momentum Is Always Conserved in Isolated Collisions
During a collision, interaction forces between bodies are internal to the chosen system and occur in equal-and-opposite pairs. That is why total momentum remains fixed for the system even while individual body momenta can change dramatically.
What Kinetic Energy Tells You
Collision type is determined by kinetic-energy behavior. In an elastic collision, kinetic energy is conserved in addition to momentum. In an inelastic collision, kinetic energy decreases because some is transformed to thermal energy, sound, and deformation. In a totally inelastic collision, bodies move together afterward.
This form is often useful after momentum equations are solved.
Explosions are the opposite energy direction: momentum is still conserved for the isolated system, but kinetic energy rises because internal stored energy is converted into motion.
Coefficient of Restitution as a Continuum
A practical way to model one-dimensional collisions is by restitution e, where e = 1 is perfectly elastic and e = 0 is perfectly inelastic. Real impacts typically lie in between. Combining momentum conservation with restitution predicts post-collision velocities from pre-collision states.
Relative speed of separation divided by relative speed of approach in 1D.
This ratio helps interpret outcomes quickly: e close to 1 means rebound is strong and kinetic-energy loss is small, while e close to 0 means the bodies emerge with nearly the same velocity, signaling large kinetic-energy transfer into deformation, heat, and sound.
Guided Lab: Three Collision Signatures
Use the next three simulation presets in order and record momentum and energy readouts. The goal is to build visual pattern recognition: same total momentum in every run, but systematically different kinetic-energy behavior.
Simulation: Two-Mass Collision Lab
Adjust masses, initial velocities, and elasticity, then compare initial and final momentum and kinetic energy.
Momentum Collision Lab
Collision is shown directly: two bodies move, contact, then leave with post-collision velocities from conservation + restitution.
v1 final
+1.00 m/s
v2 final
+5.00 m/s
p initial
+7.00 N·s
p final
+7.00 N·s
K initial
16.50 J
K final
13.50 J
Momentum difference: +0.0000 N·s
Energy ratio (final/initial): 0.818
Momentum is conserved in isolated collisions; kinetic energy tracks elasticity.
Interpretation: this partially inelastic case should keep momentum nearly unchanged while showing a moderate drop in kinetic energy. Check that both final velocities move toward each other compared with the approach speeds.
Simulation: Totally Inelastic Limit (e = 0)
Run a strongly uneven-mass impact where bodies should leave with a shared velocity trend characteristic of maximum kinetic-energy loss.
Momentum Collision Lab
Collision is shown directly: two bodies move, contact, then leave with post-collision velocities from conservation + restitution.
v1 final
+1.67 m/s
v2 final
+1.67 m/s
p initial
+10.00 N·s
p final
+10.00 N·s
K initial
50.00 J
K final
8.33 J
Momentum difference: +0.0000 N·s
Energy ratio (final/initial): 0.167
Momentum is conserved in isolated collisions; kinetic energy tracks elasticity.
Interpretation: as e approaches zero, post-collision speeds converge. Momentum is still conserved, but kinetic-energy loss is largest for the same initial momenta. This links directly to textbook 'stick together' style examples.
Simulation: Near-Elastic Head-On Rebound
Set equal masses with opposite initial velocities and high elasticity to visualize a near-exchange of velocities.
Momentum Collision Lab
Collision is shown directly: two bodies move, contact, then leave with post-collision velocities from conservation + restitution.
v1 final
-1.92 m/s
v2 final
+5.92 m/s
p initial
+8.00 N·s
p final
+8.00 N·s
K initial
40.00 J
K final
38.73 J
Momentum difference: +0.0000 N·s
Energy ratio (final/initial): 0.968
Momentum is conserved in isolated collisions; kinetic energy tracks elasticity.
Interpretation: with equal masses and e near 1, velocities almost swap, and the energy ratio should stay close to 1.000. This is a useful benchmark to test sign conventions and catch algebra mistakes in worked solutions.
Test Yourself
In a 1D isolated collision, which observation most strongly indicates the collision is close to totally inelastic?
Modeling note
Use the simulation to verify a habit: momentum totals should match closely before and after for every elasticity value. Energy totals match only when elasticity is 1.