Elastic Collision

An elastic collision is a collision during which the total kinetic energy of the system is conserved: the bodies deform during contact, then fully recover their shape, releasing all the stored energy. This criterion is what defines an elastic collision.

Discover FizziQ

How to measure it in class

You do not measure the “elasticity” of a collision: you measure velocities before and after, then compare the momentum and kinetic energy balances. FizziQ offers two approaches: video analysis and acoustic timing.

Steps (video analysis, two colliding objects):

Film the collision flat, with a fixed camera perpendicular to the plane of motion, and a graduated ruler in the frame for scale
Import the video into FizziQ’s kinematic analysis module and track each object frame by frame
Read the velocities v₁ and v₂ just before contact, then v₁′ and v₂′ just after, using only a few frames on either side of the impact, since friction already changes the velocities beyond that
Weigh the two objects, then compute p before and p after: the sum m₁v₁ + m₂v₂ must be conserved, signs included
Then compute Ec before and after with ½mv², and form the ratio Ec(after)/Ec(before)
Conclude: close to 1, the collision is nearly elastic; clearly below 1, it is inelastic

Steps (acoustic timing of a bouncing ball):

Place the smartphone on the floor near the impact point and start the audio recording
Drop the ball with no initial velocity and let it bounce five to six times
Locate the instants of the successive impacts on the signal
Compute the coefficient of restitution with e = t(n+1) / t(n), the ratio of two consecutive flight intervals

Scientific activities on this topic

Elastic collision: billiards - analyze on video the collision of two billiard balls and verify both conservation laws
Elastic shock: Newton’s cradle - test the elastic collision hypothesis on a chain of balls and estimate the coefficient of restitution

Learn more

Two equations, two unknowns

In a one-dimensional elastic collision, two relations are available: conservation of momentum and conservation of kinetic energy. Two equations for two unknown final velocities: the problem is fully determined. This is what makes the model so powerful - knowing the masses and initial velocities, you predict the final state exactly, without knowing anything about the forces at play during contact.

The case of equal masses

If m₁ = m₂ and the second ball is at rest, the solution is remarkable: the incoming ball stops dead and the second one moves off at the speed the first one had. There is an exchange of velocities. This is what you observe in billiards on a head-on shot, and it is the principle of Newton’s cradle. This result is not obvious: conservation of momentum alone would allow both balls to move off together at v/2. It is the simultaneous conservation of kinetic energy that forbids it.

A property independent of the reference frame

The conservation of kinetic energy in an elastic collision holds in any inertial reference frame, even though the values of Ec themselves change from one frame to another. A useful consequence concerns the relative velocity: in an elastic collision, the relative velocity of separation after the collision is exactly the opposite of the relative velocity of approach before it. In other words, the coefficient of restitution is e = 1.

Nothing is perfectly elastic on the macroscopic scale

The perfectly elastic collision is a limiting model. Two steel balls reach e ≈ 0.90 to 0.95, two billiard balls e ≈ 0.92 to 0.96, a tennis ball on a hard floor only e ≈ 0.75. The deficit goes into heat, into vibrations of the material and into sound - the “clack” of the impact is lost kinetic energy, made audible. The only truly elastic collisions are microscopic: collisions between molecules of an ideal gas, elastic scattering of a neutron on a nucleus. An ideal gas is in fact partly defined by this assumption: without it, the pressure of an enclosed gas would collapse within moments.

In the curriculum

Collisions are covered in the final years of French high school physics and chemistry, in the study of interacting systems, the kinetic energy theorem, and Newton’s second law in momentum form.

Formula

Conservation of momentum (valid in all collisions of an isolated system):

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

Conservation of kinetic energy (valid only in elastic collisions):

½m₁v₁² + ½m₂v₂² = ½m₁v₁′² + ½m₂v₂′²

Final velocities, one-dimensional elastic collision:

v₁′ = [(m₁ − m₂)v₁ + 2m₂v₂] / (m₁ + m₂)

v₂′ = [(m₂ − m₁)v₂ + 2m₁v₁] / (m₁ + m₂)

Coefficient of restitution:

e = |v₂′ − v₁′| / |v₁ − v₂|, with e = 1 for an elastic collision

where:

m₁, m₂: masses of the two bodies (kg)
v₁, v₂: velocities before the collision, signed along an oriented axis (m·s⁻¹)
v₁′, v₂′: velocities after the collision (m·s⁻¹)
Ec: kinetic energy (J)
e: coefficient of restitution (dimensionless)

Application examples

Billiards, head-on collision between identical balls. A ball launched at 2.0 m·s⁻¹ at a stationary ball stops and transfers all its speed: p and Ec are conserved. With e ≈ 0.94, the incoming ball actually keeps a residual speed of a few centimeters per second.
Light ball against heavy ball. A 20 g ball hitting a stationary 200 g ball at 3 m·s⁻¹ bounces back at about 2.45 m·s⁻¹: it rebounds at nearly the same speed, the big ball acquiring only 0.55 m·s⁻¹. This is the limiting case of a ball bouncing off a wall.
Ideal gas. Air molecules collide billions of times per second. If these collisions were not elastic, the average kinetic energy - and therefore the temperature - would drop spontaneously, which is not observed.
Moderator of a nuclear reactor. Neutrons are slowed down by elastic collisions with nuclei of comparable mass, hydrogen or deuterium: this is the case m₁ ≈ m₂, the one that transfers the most energy per collision. On a heavy lead nucleus, the neutron would bounce back with almost no slowing.
Gravitational assist. A probe skimming past a planet undergoes the equivalent of an elastic collision at a distance: in the heliocentric reference frame, it leaves faster, at the cost of an infinitesimal slowing of the planet.

FAQ

Q: How can you tell whether a collision is elastic without doing any calculation? A: There is no reliable visual criterion, but two useful clues. If there is a loud impact noise, permanent deformation or heating, part of the energy has been lost. And if the two bodies leave stuck together, the collision is certainly inelastic. Only the calculation of Ec before and after settles the question.

Q: Is momentum conserved only in elastic collisions? A: No, this is the most frequent mistake. Momentum is conserved in all collisions, elastic or not, provided the external forces are negligible during the very brief contact. What distinguishes the elastic collision is the additional conservation of kinetic energy.

Q: Why can’t the two balls move off together at v/2? A: That solution does conserve momentum, but not kinetic energy: two masses m at v/2 give 2 × ½m(v/2)² = ¼mv², half of the initial energy ½mv². Half the energy is missing, which is forbidden in an elastic collision.

Q: Does a perfect elastic collision really exist? A: Not at our scale. The best steel balls reach e ≈ 0.95, which still leaves about 10% of the kinetic energy dissipated at each collision, since the energy ratio is e². Truly elastic collisions are those of particles and molecules.

Q: Can an elastic collision make objects spin? A: Yes, and it is common as soon as the collision is not head-on. Rotational kinetic energy then counts in the balance: an object that leaves spinning has indeed conserved its total energy, but not its translational energy alone. In billiards, this is the whole art of spin.

Inelastic Collision - Momentum - Kinetic Energy - Conservation of Energy - Coefficient of Restitution - Newton’s Cradle - Elastic Energy - Inertial Reference Frame