Difficulty: Introductory | Prerequisites: Momentum and conservation of momentum (see Part 1 notes).
Once you know that momentum is conserved in collisions, the next questions are: how fast do the objects separate, and does the system lose any energy? This topic introduces approach and separation velocity (how quickly objects close in and fly apart), kinetic energy (the energy of motion) and elasticity (how much bounce a collision retains). Together these ideas let you fully predict the outcome of any two-body collision, which is the endgame of this unit. You should already be comfortable with p = mv and the sign convention for velocity.
In a perfectly elastic collision (elasticity = 1.0), the approach velocity equals the separation velocity and total kinetic energy is conserved. When elasticity drops below 1.0, some kinetic energy converts to heat or sound, and the separation velocity shrinks proportionally. The ratio of separation velocity to approach velocity gives you the elasticity of the collision.
Approach velocity
The relative speed at which two objects move toward each other before a collision: v_approach = v1 − v2. Think of it as how fast the gap between them is closing.
Separation velocity
The relative speed at which two objects move apart after a collision: v_separation = v2' − v1'. In simple terms, it measures how fast the gap between them opens up after impact.
Kinetic energy (KE)
The energy an object possesses because of its motion, measured in joules (J): KE = mv²/2. Think of it as the energy you would have to supply (or absorb) to bring the object from rest to that speed.
Conservation of energy
In a closed system, the total energy cannot be created or destroyed, only converted from one form to another. In a perfectly elastic collision, all kinetic energy is conserved. In an inelastic one, some converts to heat or sound.
Elasticity (coefficient of restitution)
A number between 0 and 1 that describes how "bouncy" a collision is. It equals the ratio of separation velocity to approach velocity: e = v_separation / v_approach. An elasticity of 1.0 means a perfectly elastic collision; 0 means the objects stick together (perfectly inelastic).
Elastic collision
A collision in which both momentum and kinetic energy are conserved. The objects bounce off each other with no energy lost to deformation, heat or sound.
Inelastic collision
A collision in which momentum is conserved but some kinetic energy is converted to other forms. The objects may deform, produce sound or generate heat.
Before a collision, the approach velocity tells you how fast the two objects are closing in: v_approach = v1 − v2.
After a collision, the separation velocity tells you how fast they move apart: v_separation = v2' − v1'.
In a perfectly elastic collision (elasticity = 1.0), the approach velocity equals the separation velocity every time.
Example: m1 = 3.0 kg at 4.0 m/s, m2 = 1.5 kg at −6.0 m/s. Approach velocity = 4.0 − (−6.0) = 10.0 m/s. After the collision, v1' = −2.67 m/s, v2' = 7.33 m/s. Separation velocity = 7.33 − (−2.67) = 10.0 m/s.
With two equations (conservation of momentum and approach = separation velocity), you can solve for both final velocities algebraically.
Set up: m1·v1 + m2·v2 = m1·v1' + m2·v2' and v1 − v2 = v2' − v1'.
Example: m1 = 2.0 kg, v1 = 2.6 m/s, m2 = 3.0 kg, v2 = −4.4 m/s. Solving gives v1' = −5.80 m/s, v2' = 1.20 m/s.
Kinetic energy is calculated as KE = mv²/2. Note: because velocity is squared, KE is always positive (or zero).
In a perfectly elastic collision, the total KE before equals the total KE after.
Example: m1 = 2.4 kg at 4.40 m/s, m2 = 2.4 kg at −4.40 m/s. Each has KE = 23.23 J. Total = 46.46 J. After the collision, the gliders swap velocities and the total KE remains 46.46 J.
When elasticity is less than 1.0, some kinetic energy is lost to heat, sound or deformation during the collision.
The elasticity equals the ratio of separation velocity to approach velocity: e = v_separation / v_approach.
Example data at different elasticity settings (same approach velocity of 10.6 m/s): at e = 0.2, separation velocity = 2.12 m/s; at e = 0.6, separation velocity = 6.36 m/s; at e = 0.8, separation velocity = 8.48 m/s.
The lower the elasticity, the more kinetic energy the system loses. At e = 0, the objects stick together (perfectly inelastic collision).
Momentum is still conserved regardless of the elasticity value. Only kinetic energy changes.
v_{\text{approach}} = v_1 - v_2Approach velocity: the rate at which two objects close in before colliding.
v_{\text{separation}} = v_2' - v_1'Separation velocity: the rate at which two objects move apart after colliding.
KE = \frac{1}{2}mv^2Kinetic energy of a single object.
e = \frac{v_{\text{separation}}}{v_{\text{approach}}}Elasticity (coefficient of restitution): the ratio of separation to approach velocity. Ranges from 0 (perfectly inelastic) to 1 (perfectly elastic).
Elasticity determines how sports equipment performs: a tennis ball is designed with high elasticity so it bounces well off the racket, while a cricket ball's lower elasticity means more energy goes into deforming the ball and producing that distinctive sound. Vehicle crash testing uses the same principles: crumple zones deliberately lower the elasticity of a collision, absorbing kinetic energy as deformation to protect passengers.
Students often think kinetic energy can be negative. It cannot. Because velocity is squared in the formula, KE is always zero or positive, regardless of which direction the object moves.
Students sometimes assume kinetic energy is always conserved in collisions. It is conserved only in perfectly elastic collisions (elasticity = 1.0). In all other collisions, some KE is converted to heat, sound or permanent deformation.
Students frequently confuse conservation of momentum with conservation of kinetic energy. Momentum is always conserved in a closed-system collision. Kinetic energy is not, unless the collision is perfectly elastic.
Students sometimes think a low elasticity means momentum is lost. Momentum is still fully conserved at any elasticity value. Only kinetic energy changes.
Expect problems that ask you to calculate approach and separation velocities and then determine the elasticity.
Be prepared to solve for unknown final velocities using two simultaneous equations: conservation of momentum and the approach-equals-separation condition.
Know the difference between elastic and inelastic collisions cold. A common exam question gives you before-and-after KE values and asks whether energy was conserved.
Watch for questions that give an elasticity value below 1.0 and ask you to find the separation velocity or the energy lost.
True or false: kinetic energy can be negative. (False. Velocity is squared, so KE is always zero or positive.)
Fill in the blank: elasticity = ______ / ______. (separation velocity / approach velocity)
True or false: in a perfectly elastic collision, the approach velocity equals the separation velocity. (True.)
An object of mass 2 kg moves at 5 m/s. Its kinetic energy is ______ J. (25)
True or false: momentum is conserved in an inelastic collision. (True. Momentum is always conserved in a closed system, regardless of elasticity.)
Q: Glider 1 (3.0 kg) moves at 4.0 m/s. Glider 2 (1.5 kg) moves at −6.0 m/s. What is the approach velocity?
A: v_approach = 4.0 − (−6.0) = 10.0 m/s.
Q: After the collision above, Glider 1 has velocity −2.67 m/s and Glider 2 has velocity 7.33 m/s. What is the separation velocity, and what does it tell you about the collision?
A: v_separation = 7.33 − (−2.67) = 10.0 m/s. Because the separation velocity equals the approach velocity, the collision is perfectly elastic (e = 1.0).
Q: A collision has an approach velocity of 10.6 m/s and an elasticity of 0.6. What is the separation velocity?
A: v_separation = e × v_approach = 0.6 × 10.6 = 6.36 m/s.
Q: Two gliders of mass 2.4 kg each move at 4.40 m/s and −4.40 m/s. Calculate the total kinetic energy before the collision.
A: KE1 = 0.5 × 2.4 × 4.40² = 23.23 J. KE2 = 0.5 × 2.4 × 4.40² = 23.23 J. Total KE = 46.46 J.
Q: After the collision above, the gliders swap velocities. Is kinetic energy conserved? Explain.
A: Yes. Each glider now moves at the speed the other had before, so each KE is unchanged. Total KE after = 46.46 J, the same as before. This confirms a perfectly elastic collision.
Elasticity and kinetic energy connect directly to the broader principle of energy conservation, which you will revisit when studying work, potential energy and thermodynamics. The coefficient of restitution reappears in engineering contexts such as materials science, where it characterises how different materials absorb impact energy. Approach and separation velocities also lay the groundwork for understanding relative motion in frames of reference, which becomes central in later mechanics and in introductory relativity.
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