Collisions: Elastic, Inelastic & In Between
Momentum always survives a collision; kinetic energy only sometimes.
Lesson at a glanceShow the key points
- Total momentum is conserved throughout a collision when external forces are negligible.
- During contact, kinetic energy dips as deformation energy is stored; maximum compression occurs when the bodies share a common velocity.
- Elastic: final KE = initial KE. Perfectly inelastic: bodies stick together, V = (m₁v₁ + m₂v₂)/(m₁ + m₂). Partially elastic: in between.
- KE lost in a perfectly inelastic collision = m₁m₂(v₁ − v₂)² / [2(m₁ + m₂)].
- Coefficient of restitution: velocity of separation = e × velocity of approach; e = 1 elastic, e = 0 perfectly inelastic.
Two bodies meet, push hard on each other for a moment, and separate (or stick). We rarely know the forces during that moment, but momentum lets us connect “before” and “after” anyway.
What happens inside a collision
Picture a block sliding toward an identical block at rest, with a light spring bumper on the front of the second one. When the spring touches, it compresses: it slows the first block and speeds up the second. While block 1 is still faster, the gap keeps closing and the spring squeezes harder. Compression is greatest at the instant both blocks move with the same velocity. After that the spring pushes them apart until it regains its natural length.
Throughout, the spring forces are internal, so momentum is conserved at every instant. Kinetic energy is not: at maximum compression some of it is stored in the spring. Whether all of it comes back decides the type of collision.
A 2.0 kg block moving at 3.0 m/s runs into an identical block at rest. The second block carries a light spring of stiffness 600 N/m on its front. Find the maximum compression (no friction).
Elastic and inelastic
A 60 kg trolley rolls east at 18 km/h and a 40 kg trolley rolls west at 12 km/h on the same track. They collide and their couplings lock. Find their common velocity.
For any perfectly inelastic collision on a line, a little algebra gives the kinetic energy lost:
Coefficient of restitution
Real collisions fall between the two extremes. Measure where with the ratio of the speed of separation after the collision to the speed of approach before it:
Combine it with momentum conservation and the final velocities follow for any masses and any e:
A puck of mass m sliding at speed v hits a puck of mass 3m at rest, head-on, and stops dead. Find e.
- 1Momentum is conserved in every collision; kinetic energy only in elastic ones.
- 2Maximum deformation happens when the bodies share a common velocity.
- 3Perfectly inelastic: bodies stick; maximum possible KE loss.
- 4 = separation speed ÷ approach speed, from 0 (stick) to 1 (elastic).
Check yourself
Keep exploring
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