C3.3 · Distinguish elastic and inelastic collisions

Ontario Grade 12 Physics

Energy and Momentum

Classifying collisions by comparing kinetic energy

What you will learn

• Define a collision system, reference frame, and positive direction.
• Distinguish elastic and inelastic collisions by comparing total kinetic energy before and after.
• Calculate the total kinetic energy of a system of colliding objects.
• Explain why a decrease in kinetic energy does not mean energy has disappeared.

1. Prerequisite bridge: motion, mass, and energy

• Velocity is a vector; kinetic energy is a scalar.
• Use kilograms, metres per second, and joules.
• Keep the system and reference frame consistent.

2. The defining difference

• Elastic: total kinetic energy before equals total kinetic energy after.
• Inelastic: total kinetic energy after is less than before.
• Perfectly inelastic: the objects stick together after impact.
• Compare the total for the system, not one object by itself.

3. A reliable classification method

• Square the velocity, including when it is negative.
• Add the kinetic energies of all objects in the system.
• Use the before-and-after totals to classify the collision.

Compare collision types

Worked example

1. Total kinetic energy stays the same

1. Set the system and direction
   Take both carts as the system and the track as the reference frame. Right is positive. The masses and velocities are known; the unknown is the collision type.
2. Find the initial total
   Calculate the kinetic energy of each cart and add the values. Cart B starts at rest, so its initial kinetic energy is zero.
   $$E_{k,\mathrm{before}}=\frac{1}{2}(0.50\ \mathrm{kg})(2.0\ \mathrm{m/s})^2+\frac{1}{2}(0.50\ \mathrm{kg})(0\ \mathrm{m/s})^2=1.0\ \mathrm{J}$$
3. Find the final total
   After impact, A is at rest and B moves at the stated velocity. Add their final kinetic energies.
   $$E_{k,\mathrm{after}}=\frac{1}{2}(0.50\ \mathrm{kg})(0\ \mathrm{m/s})^2+\frac{1}{2}(0.50\ \mathrm{kg})(2.0\ \mathrm{m/s})^2=1.0\ \mathrm{J}$$

Worked example

2. Objects stick together

1. Calculate the initial energy
   The first cart is moving and the second is at rest. Add their initial kinetic energies. The positive sign shows direction, while the squared velocity determines the kinetic energy.
   $$E_{k,\mathrm{before}}=\frac{1}{2}(1.0\ \mathrm{kg})(3.0\ \mathrm{m/s})^2+\frac{1}{2}(2.0\ \mathrm{kg})(0\ \mathrm{m/s})^2=4.5\ \mathrm{J}$$
2. Calculate the final energy
   The carts move together after impact. Their combined mass is $3.0\ \mathrm{kg}$ and their shared velocity is $+1.0\ \mathrm{m/s}$. Use these values to find the final total.
   $$E_{k,\mathrm{after}}=\frac{1}{2}(3.0\ \mathrm{kg})(1.0\ \mathrm{m/s})^2=1.5\ \mathrm{J}$$
3. Compare the totals
   Subtract the final total from the initial total. The objects stick together and the total is smaller afterward, so this is an inelastic collision.
   $$\Delta E_k=4.5\ \mathrm{J}-1.5\ \mathrm{J}=3.0\ \mathrm{J}$$

Worked example

3. Direction changes, but the energy test stays the same

1. Calculate the initial total
   Use both objects as the system. The negative velocity means A initially moves opposite to the positive direction. Squaring that velocity still gives a positive kinetic energy.
   $$E_{k,\mathrm{before}}=\frac{1}{2}(0.20\ \mathrm{kg})(-4.0\ \mathrm{m/s})^2+\frac{1}{2}(0.40\ \mathrm{kg})(2.0\ \mathrm{m/s})^2=2.4\ \mathrm{J}$$
2. Calculate the final total
   Use each final velocity squared, then add the kinetic energies. The negative velocity for B describes its direction, not a negative energy.
   $$E_{k,\mathrm{after}}=\frac{1}{2}(0.20\ \mathrm{kg})(2.0\ \mathrm{m/s})^2+\frac{1}{2}(0.40\ \mathrm{kg})(-1.0\ \mathrm{m/s})^2=0.60\ \mathrm{J}$$
3. Classify the collision
   Compare the totals. The final total is smaller, so the collision is inelastic. A change in direction does not alter the classification rule.
   $$0.60\ \mathrm{J}<2.4\ \mathrm{J}$$

Common mistakes and how to avoid them

Lesson summary

• Define the system, reference frame, and positive direction before describing a collision.
• Kinetic energy is a scalar calculated from mass and velocity squared.
• An elastic collision has equal total kinetic energy before and after.
• An inelastic collision has less total kinetic energy after than before.
• In a perfectly inelastic collision, the objects stick together.

Check your understanding

Key terms

Collision

A brief interaction in which objects exert forces on one another and their motion can change.

System

The object or group of objects chosen for analysis.

Reference frame

The viewpoint and coordinate system used to describe position and motion.

Elastic collision

A collision in which the system's total kinetic energy is the same before and after.

Inelastic collision

A collision in which the system's total kinetic energy is lower after than before.

Perfectly inelastic collision

An inelastic collision in which the colliding objects stick together after impact.

Continue through SPH4U

View the complete SPH4U Ontario Grade 12 Physics curriculum and lessons

• C1.1 · Analyse and improve a technology using energy and momentum
• C1.2 · Assess impacts of energy- and momentum-based technologies
• C2.1 · Use work, energy, impulse, momentum, and collision terminology
• C2.2 · Solve one- and two-dimensional work-energy problems
• C2.3 · Analyse mechanical and thermal energy systems through inquiry
• C2.4 · Test conservation of energy during transformations

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C3.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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