C2.7 Test conservation laws with collisions and explosions

Ontario Grade 12 Physics

Energy and Momentum

Using momentum and kinetic energy to compare before-and-after motion

In SPH3U, you learned that velocity has magnitude and direction, while speed has magnitude only. Momentum also has direction: it is a vector. Kinetic energy is a scalar, so it has magnitude but no direction. In this lesson, you will use these ideas to test what happens during collisions and explosions. A test compares values before and after an interaction. A calculation can predict what should happen, but it is not the same as collecting experimental measurements.

What you will learn

Define a system, reference frame, and positive direction for an interaction.
Use momentum conservation to predict motion after a collision or explosion.
Compare kinetic energy before and after to classify a collision.
Describe how measured evidence can test whether a conservation law is supported.

1. Choose the system and direction

A system is the object or group of objects being studied. For a collision, the system often includes both colliding objects. For an explosion, it includes all the pieces that move apart. Choose the system before applying a conservation law.

A reference frame is the viewpoint used to describe position and motion. For typical classroom collision investigations, use the room or track as the reference frame. Choose one direction along the track as positive. Motion in the opposite direction is negative. Keep this choice for every velocity in the calculation.

Momentum depends on mass and velocity. Its SI unit is the kilogram metre per second. Kinetic energy depends on mass and speed squared, and its SI unit is the joule. Since momentum is a vector, signs or components show direction. Kinetic energy is never negative.

p=mv

Define the system, reference frame, and positive direction before calculating.
Momentum is a vector; kinetic energy is a scalar.

2. Apply momentum conservation

The total momentum of a system is the vector sum of the momentum of every object in it. During a brief collision or explosion, momentum is conserved when the system is isolated, meaning external forces have little or no net effect during the interaction. Friction or a push from outside the chosen system can affect the result.

For motion along one straight line, use signed velocities. A velocity to the left is negative if right was chosen as positive. In two dimensions, compare horizontal and vertical components separately. Do not add momentum magnitudes when the objects move in different directions.

An explosion is an interaction in which parts of a system move apart. If the system was initially at rest and external effects are negligible, its initial total momentum is zero. The pieces must then have momenta that balance as vectors. This does not mean their speeds must be equal; their masses matter.

\sum p_{\mathrm{before}}=\sum p_{\mathrm{after}}

Use signed quantities for one-dimensional motion.
Include every object in the system before and after the interaction.
Momentum conservation predicts the total after-interaction momentum, not necessarily the motion of each object by itself.

3. Use kinetic energy to describe the collision

A collision can conserve momentum while the objects' total kinetic energy changes. An elastic collision is one in which the system's total kinetic energy before and after is the same. In an inelastic collision, some kinetic energy is transformed into other forms, such as sound, heating, or deformation. If objects stick together, the collision is perfectly inelastic.

Kinetic energy is calculated for each moving object and then added. It is not a vector, so direction does not give it a negative sign. Compare the total for the chosen system before and after, using the same set of objects.

In a real investigation, measurements will not be perfectly exact. Small differences may come from measurement limits or external effects. Record the measured masses and velocities, calculate totals, and compare the results. A close match supports the model within the limits of the measurements; it does not prove that every possible interaction follows the model.

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

Momentum and kinetic energy are different quantities.
Momentum can be conserved even when kinetic energy is not.
Use measured evidence for a claim about an actual investigation; label calculated or simulated results clearly.

4. A fair test of conservation

A collision investigation can use carts on a track. Measure each cart's mass and determine its velocity just before and just after the interaction. A motion sensor or video analysis may be used to estimate velocity, depending on the available equipment. This is a proposed procedure, not a report of collected results.

For each trial, calculate the total momentum before and after. Then calculate the total kinetic energy before and after. Keep units, signs, and the chosen positive direction consistent. Repeat trials if possible, and note limitations such as friction or uncertainty in the velocity readings.

For an explosion investigation, begin with connected objects at rest, then let them separate. Measure the mass and velocity of every piece. Test whether their vector momenta add to the initial momentum. If the pieces move in different directions, resolve their momenta into components. A simulation can help practise these calculations, but simulated output is not measured laboratory evidence.

\Delta p=p_{\mathrm{after}}-p_{\mathrm{before}}

Record actual measurements separately from predictions or simulated values.
Compare totals for the same system, and state factors that may affect the comparison.
A reasonable test checks both the numerical result and the direction of each momentum.

Worked example

A cart collision where the carts stick

Cart A has mass

0.80\,\mathrm{kg}

and moves right at

2.0\,\mathrm{m/s}

. Cart B has mass

1.20\,\mathrm{kg}

and is at rest. They stick together. Find their final velocity and compare the total kinetic energies.

Set the system and direction
Use both carts as the system and the track as the reference frame. Choose right as positive. The unknown is the shared final velocity, $v_f$ .
Use momentum conservation
Treat the interaction as isolated along the track. Because the carts stick, they have the same final velocity. Substitute the masses in kilograms and velocities in metres per second.
$(0.80\,\mathrm{kg})(+2.0\,\mathrm{m/s})+(1.20\,\mathrm{kg})(0\,\mathrm{m/s})=(2.00\,\mathrm{kg})v_f$
Solve for the velocity
Divide the initial momentum by the combined mass. The positive sign means the carts move right.
$v_f=+0.80\,\mathrm{m/s}$
Compare kinetic energy
The initial kinetic energy is from cart A only. Afterward, both carts move together. The decrease is consistent with a perfectly inelastic collision.
$E_{k,i}=1.6\,\mathrm{J},\qquad E_{k,f}=0.64\,\mathrm{J}$

Answer: The carts move together at

0.80\,\mathrm{m/s}

to the right. Momentum is conserved in the model, while total kinetic energy decreases from

1.6\,\mathrm{J}

to

0.64\,\mathrm{J}

.

Check: Momentum before and after is

1.6\,\mathrm{kg\,m/s}

to the right. The final speed is below the initial speed, which is reasonable because the carts combine into a larger mass. Units are consistent.

Worked example

An explosion from rest

A two-part object with total mass

3.0\,\mathrm{kg}

is initially at rest. It separates into a

1.0\,\mathrm{kg}

piece moving right at

6.0\,\mathrm{m/s}

and a

2.0\,\mathrm{kg}

piece. Find the second piece's velocity.

Define the system
Include both pieces in the system and use the ground as the reference frame. Choose right as positive. The initial total momentum is zero because the object is at rest.
$p_i=(3.0\,\mathrm{kg})(0\,\mathrm{m/s})=0\,\mathrm{kg\,m/s}$
Balance the final momentum
If external effects are negligible, the two final momenta must add to zero. Let $v_2$ be the velocity of the $2.0\,\mathrm{kg}$ piece.
$(1.0\,\mathrm{kg})(+6.0\,\mathrm{m/s})+(2.0\,\mathrm{kg})v_2=0$
Solve and state direction
The second piece must have negative momentum to balance the first. A negative velocity means left under the chosen sign convention.
$v_2=-3.0\,\mathrm{m/s}$

Answer: The

2.0\,\mathrm{kg}

piece moves at

3.0\,\mathrm{m/s}

to the left.

Check: The final momenta are

+6.0\,\mathrm{kg\,m/s}

and

-6.0\,\mathrm{kg\,m/s}

, giving zero total momentum. The heavier piece has the lower speed, as expected for equal and opposite momenta.

Worked example

Testing a proposed elastic collision

A

0.50\,\mathrm{kg}

cart moving right at

2.0\,\mathrm{m/s}

collides with a stationary

0.50\,\mathrm{kg}

cart. A calculation predicts that the first cart stops and the second moves right at

2.0\,\mathrm{m/s}

. Test momentum and kinetic energy for this prediction.

Set the frame
Use both carts as the system, the track as the reference frame, and right as positive. Treat the given values as a prediction to test, not as measured experimental results.
Compare momentum
Add the signed momenta before and after. The same total supports momentum conservation for this model.
$p_i=(0.50)(+2.0)=+1.0\,\mathrm{kg\,m/s},\qquad p_f=(0.50)(0)+(0.50)(+2.0)=+1.0\,\mathrm{kg\,m/s}$
Compare kinetic energy
Calculate the total kinetic energy on each side. Both totals match, so the prediction is consistent with an elastic collision.
$E_{k,i}=1.0\,\mathrm{J},\qquad E_{k,f}=1.0\,\mathrm{J}$

Answer: The prediction conserves both total momentum and total kinetic energy, so it is consistent with an elastic collision.

Check: Momentum has units of

\mathrm{kg\,m/s}

and kinetic energy has units of joules. This calculation tests the prediction; actual measured values would be needed to test a real collision.

Common mistakes and how to avoid them

Adding momentum magnitudes even when objects move in opposite directions.

Correction: Choose a positive direction and use signed velocities, or compare vector components.

Assuming kinetic energy is conserved in every collision.

Correction: Test kinetic energy separately. Momentum may be conserved while kinetic energy changes.

Leaving an object out of the system after an explosion or collision.

Correction: Include every interacting object or piece in the before-and-after total.

Calling a calculated prediction measured evidence.

Correction: Identify whether values are measured, calculated, or simulated, and do not claim an investigation was performed unless data were collected.

Lesson summary

Define the system, reference frame, and positive direction before solving.
For an isolated system, compare total vector momentum before and after an interaction.
Compare total kinetic energy separately to identify whether a collision is elastic or inelastic.
Use measurements to test a real event and discuss limitations in the evidence.

Check your understanding

Question 1

Two objects have equal and opposite momentum before an explosion. What is their total momentum?

Zero
Twice the magnitude of either momentum
The momentum of the heavier object only
It cannot be determined without their kinetic energies

Show answer and explanation

Zero

Equal and opposite vectors add to zero. If external effects are negligible, the total momentum after the explosion must also be zero.

Question 2

A collision has the same total momentum before and after, but less total kinetic energy afterward. Which statement fits?

Momentum conservation failed.
The collision is inelastic.
Kinetic energy is a vector.
The objects must have stuck together.

Show answer and explanation

The collision is inelastic.

An inelastic collision has a decrease in total kinetic energy, even when momentum is conserved. The objects need not stick together.

Question 3

A

0.20\,\mathrm{kg}

object initially at rest separates into two pieces. One piece has momentum

+0.30\,\mathrm{kg\,m/s}

. If the system is isolated, what is the other piece's momentum?

$+0.30\,\mathrm{kg\,m/s}$
$-0.30\,\mathrm{kg\,m/s}$
$0\,\mathrm{kg\,m/s}$
$+0.60\,\mathrm{kg\,m/s}$

Show answer and explanation

-0.30\,\mathrm{kg\,m/s}

Initial total momentum is zero. The second piece must have equal momentum in the opposite direction so the final vector sum remains zero.

Key terms

System: The object or group of objects selected for study.
Reference frame: The viewpoint used to describe an object's motion.
Momentum: A vector quantity equal to an object's mass multiplied by its velocity.
Isolated system: A system with negligible net external effect during the interaction being studied.
Elastic collision: A collision in which total kinetic energy before and after is the same.
Inelastic collision: A collision in which total kinetic energy decreases.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C2.7. 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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C2.7 · Test conservation laws with collisions and explosions

Ontario Grade 12 Physics

Using momentum and kinetic energy to compare before-and-after motion

What you will learn

1. Choose the system and direction

2. Apply momentum conservation

3. Use kinetic energy to describe the collision

4. A fair test of conservation

A cart collision where the carts stick

An explosion from rest

Testing a proposed elastic collision

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review