Distinguish elastic and inelastic collisions

Ontario Grade 12 Physics

Energy and Momentum

Classifying collisions by comparing kinetic energy

A collision is a brief interaction in which objects exert forces on one another and their motion can change. A ball hitting another ball and two carts bumping are examples. Before describing a collision, choose the system, reference frame, and positive direction. In this lesson, the system is the colliding objects together, the reference frame is the track or floor, and right is positive. Velocity is a vector: it has magnitude and direction. Kinetic energy is a scalar: it has magnitude but no direction. The key distinction is whether the system’s total kinetic energy is the same before and after the collision.

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

A collision can change an object's velocity. Velocity describes how quickly position changes and in which direction. In one dimension, a velocity to the right is positive under our chosen convention; a velocity to the left is negative. Mass is measured in kilograms, and velocity is measured in metres per second.

Kinetic energy is the energy an object has because it is moving. It depends on mass and the square of speed. Speed is the magnitude of velocity, so squaring velocity gives a non-negative value. Kinetic energy is measured in joules. Since kinetic energy is a scalar, do not attach a left or right direction to it.

Compare the same system in the same reference frame before and after the collision. For example, if two carts collide, include both carts in the system. Use the track as the reference frame for both descriptions. This makes the comparison consistent.

E_k=\frac{1}{2}mv^2

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

An elastic collision is a collision in which the system's total kinetic energy is the same before and after. To find the total, calculate the kinetic energy of each object in the system and add the values. The objects' individual kinetic energies may change. It is the system total that must remain the same.

An inelastic collision is one in which the system's total kinetic energy is lower after the collision than before. Some kinetic energy changes into other forms, such as sound, heating, or a lasting change in shape. The collision is classified by its kinetic energy, not by whether all forms of energy have stayed the same.

In a perfectly inelastic collision, the objects stick together after impact and move with a shared velocity. This is one kind of inelastic collision. Not every inelastic collision makes the objects stick together.

Do not classify a collision only by whether the objects bounce apart or remain together. Objects that bounce apart can still have less total kinetic energy after the collision. Compare the system's total kinetic energy before and after.

E_{k,\mathrm{before}}=E_{k,\mathrm{after}}\quad\text{(elastic)}

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

First, list the objects in the system. Record each object's mass and velocity before and after the collision. Use the chosen positive direction consistently. A negative velocity means motion opposite to the positive direction; it does not mean negative kinetic energy.

Next, calculate each object's kinetic energy using its mass and its velocity squared. Add the objects' kinetic energies for the before total. Repeat for the after total. Compare the two totals. Equal totals indicate an elastic collision. A smaller after total indicates an inelastic collision.

Use suitable significant figures. Small differences caused only by rounding should not be treated as a meaningful change. In a classroom question, follow the precision of the given values. Check that each kinetic energy is non-negative and has units of joules. The result should make sense: changing direction does not make kinetic energy negative.

E_{k,\mathrm{total}}=\sum_i\frac{1}{2}m_i v_i^2

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

Collision type	System's total kinetic energy	What to look for
Elastic	Same before and after	Compare total kinetic energy
Inelastic	Lower after than before	Some kinetic energy changes to other forms
Perfectly inelastic	Lower after than before	Objects stick together

Worked example

1. Total kinetic energy stays the same

Two carts move on a straight track. In the track reference frame, cart A has mass

0.50\ \mathrm{kg}

and velocity

+2.0\ \mathrm{m/s}

before impact. Cart B has mass

0.50\ \mathrm{kg}

and is initially at rest. After impact, A is at rest and B moves at

+2.0\ \mathrm{m/s}

. Classify the collision.

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.
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.
$E_{k,\mathrm{before}}=\frac{1}{2}(0.50\ \mathrm{kg})(2.0\ \mathrm{m/s})^2+\frac{1}{2}(0.50\ \mathrm{kg})(0\ \mathrm{m/s})^2=1.0\ \mathrm{J}$
Find the final total
After impact, A is at rest and B moves at the stated velocity. Add their final kinetic energies.
$E_{k,\mathrm{after}}=\frac{1}{2}(0.50\ \mathrm{kg})(0\ \mathrm{m/s})^2+\frac{1}{2}(0.50\ \mathrm{kg})(2.0\ \mathrm{m/s})^2=1.0\ \mathrm{J}$

Answer: The collision is elastic because the total kinetic energy is

1.0\ \mathrm{J}

both before and after.

Check: The units reduce to

\mathrm{kg\,m^2/s^2}

, which is a joule. Both totals are positive and equal. The carts' individual motion changed, but the system's total kinetic energy did not.

Worked example

2. Objects stick together

A

1.0\ \mathrm{kg}

cart moves right at

+3.0\ \mathrm{m/s}

and collides with a stationary

2.0\ \mathrm{kg}

cart. They stick together and move right at

+1.0\ \mathrm{m/s}

. Use the carts as the system and the track as the reference frame. Classify the collision and find the decrease in total kinetic energy.

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.
$E_{k,\mathrm{before}}=\frac{1}{2}(1.0\ \mathrm{kg})(3.0\ \mathrm{m/s})^2+\frac{1}{2}(2.0\ \mathrm{kg})(0\ \mathrm{m/s})^2=4.5\ \mathrm{J}$
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.
$E_{k,\mathrm{after}}=\frac{1}{2}(3.0\ \mathrm{kg})(1.0\ \mathrm{m/s})^2=1.5\ \mathrm{J}$
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.
$\Delta E_k=4.5\ \mathrm{J}-1.5\ \mathrm{J}=3.0\ \mathrm{J}$

Answer: The collision is inelastic. The total kinetic energy decreases by

3.0\ \mathrm{J}

.

Check: The final kinetic energy is less than the initial kinetic energy, as required for an inelastic collision. Both values have units of joules. The decrease means kinetic energy changed into other forms; it did not disappear.

Worked example

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

Two objects move along a straight line. In the chosen reference frame, object A has mass

0.20\ \mathrm{kg}

and velocity

-4.0\ \mathrm{m/s}

before impact. Object B has mass

0.40\ \mathrm{kg}

and velocity

+2.0\ \mathrm{m/s}

. After impact, A moves at

+2.0\ \mathrm{m/s}

and B at

-1.0\ \mathrm{m/s}

. Classify the collision.

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.
$E_{k,\mathrm{before}}=\frac{1}{2}(0.20\ \mathrm{kg})(-4.0\ \mathrm{m/s})^2+\frac{1}{2}(0.40\ \mathrm{kg})(2.0\ \mathrm{m/s})^2=2.4\ \mathrm{J}$
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.
$E_{k,\mathrm{after}}=\frac{1}{2}(0.20\ \mathrm{kg})(2.0\ \mathrm{m/s})^2+\frac{1}{2}(0.40\ \mathrm{kg})(-1.0\ \mathrm{m/s})^2=0.60\ \mathrm{J}$
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.
$0.60\ \mathrm{J}<2.4\ \mathrm{J}$

Answer: The collision is inelastic.

Check: Each energy is non-negative and has units of joules. The after total is smaller, so the classification is consistent. Negative velocity affects the direction description, not the sign of kinetic energy.

Common mistakes and how to avoid them

Calling a collision elastic whenever the objects bounce apart.

Correction: Bouncing apart does not decide the type. Compare the system's total kinetic energy before and after.

Treating negative velocity as negative kinetic energy.

Correction: Velocity includes direction, but kinetic energy uses velocity squared and is non-negative.

Treating kinetic energy as the only form of energy.

Correction: Kinetic energy can change into sound, heating, or a change in shape. The collision classification concerns kinetic energy.

Assuming every inelastic collision makes the objects stick together.

Correction: Sticking together identifies a perfectly inelastic collision. In other inelastic collisions, the objects may move separately afterward.

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

Question 1

A system has total kinetic energy

6.0\ \mathrm{J}

before a collision and

6.0\ \mathrm{J}

after it. How is the collision classified?

Elastic
Inelastic
Perfectly inelastic in every case
Impossible to classify from the kinetic-energy totals

Show answer and explanation

Elastic

The before-and-after totals are equal, so the collision is elastic. Equal total kinetic energy is the defining test.

Question 2

An object's velocity is

-3.0\ \mathrm{m/s}

. Which statement about its kinetic energy is correct?

It is negative because the velocity is negative.
It is positive because the velocity is squared.
It is zero because the object moves in the negative direction.
It has a negative direction because velocity is a vector.

Show answer and explanation

It is positive because the velocity is squared.

Squaring the velocity gives a positive value. Kinetic energy is a scalar, so it has no direction.

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.

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

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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