C1.1 Analyse and improve a technology using energy and momentum

Ontario Grade 12 Physics

Energy and Momentum

A Grade 12 approach to analysing how designs transfer energy and change motion

Many technologies work by transferring energy, changing momentum, or controlling how quickly an object changes motion. A vehicle restraint, a sports helmet, and a catching net all manage these changes in different ways. In this lesson, a system is the object or group of objects being studied. A reference frame is the viewpoint used to describe positions and motion. Choose a positive direction before assigning signs to velocity or force. Momentum is a vector: it has magnitude and direction. Kinetic energy is a scalar: it has magnitude but no direction. Calculations can help identify what a technology already does and what a design change might improve. Example values in this lesson are hypothetical, not measured results.

What you will learn

Define a system, reference frame, and positive direction before analysing motion.
Use momentum and kinetic energy to describe what happens during an interaction.
Use evidence from calculations to suggest a practical improvement to a technology.
Check units, direction, and whether a proposed improvement is physically reasonable.

1. From motion ideas to a useful model

In SPH3U, you used velocity to describe how quickly and in what direction an object moves. You also used forces to explain changes in motion, and energy to track transfers between objects and their surroundings. These ideas support technology analysis: first identify what is moving, then decide what changes during the technology’s operation.

For a collision, the system might include both objects that interact. Choose a reference frame, such as the ground, and set a positive direction, such as to the right. State what is known and what is unknown. A velocity to the left is then negative, even though speed is positive.

A technology may reduce a force on a user without eliminating the momentum change. A cushion or crumple zone can increase the time or distance over which an object slows down. The same change in momentum spread over more time means a smaller average net force.

\vec{p}=m\vec{v}

Momentum and velocity are vectors; kinetic energy and mass are scalars.
A stated system and sign convention make calculations easier to interpret.
A proposed design change should connect to a calculated quantity, such as force or energy.

2. Conservation and technology choices

Momentum is mass multiplied by velocity. Its SI unit is the kilogram metre per second. For a system with negligible external force during a brief interaction, total momentum before the interaction equals total momentum after it. This is a model, so check whether outside forces such as friction are small enough to ignore.

Kinetic energy is the energy of motion. Its SI unit is the joule. In a collision where objects latch together, momentum can be conserved while kinetic energy decreases. The lost kinetic energy has been transferred into other forms, such as sound, deformation, or thermal energy. Energy is not destroyed.

An impulse is the change in momentum. For a constant or average net force over an interaction time, impulse equals net force multiplied by time. This relationship helps analyse protective technologies. If the momentum change is fixed, increasing the stopping time lowers the average net force.

These models do not tell a designer every detail. A real product must also be checked for factors such as fit, strength, cost, and how it is used. Calculations provide evidence about motion and energy, not a complete product test.

\sum \vec{p}_{\text{before}}=\sum \vec{p}_{\text{after}}

Use momentum conservation when external forces on the chosen system are negligible during the interaction.
Kinetic energy is not necessarily conserved in a collision, even when momentum is.
Impulse connects momentum change to average net force and interaction time.

3. Turning calculations into design improvements

A useful analysis compares the current design with a clearly stated alternative. For example, calculate the average force for one stopping time, then compare it with a longer stopping time for the same momentum change. The comparison supports a proposal, but it does not prove that a real product achieves the assumed stopping time.

Energy calculations can reveal how much motion energy a technology must manage. A larger mass or higher speed means more kinetic energy. Because speed is squared in the kinetic-energy relationship, doubling speed gives four times the kinetic energy for the same mass. This can make speed reduction an important design consideration.

Keep evidence separate from assumptions. A measured value comes from an actual measurement. A simulated or calculated value follows from a model and its inputs. In a design report, label each clearly. If proposing an investigation, describe it as a plan rather than as completed testing.

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

State which values are given, measured, assumed, or calculated.
Compare designs using the same system and consistent conditions.
A calculation supports a design proposal; it is not a substitute for testing the product.

4. A reliable analysis routine

Start by defining the system, reference frame, positive direction, and interaction. Draw a simple vector sketch if directions could be confusing. Label each object’s velocity before and after. Use signs consistently in the momentum calculation.

Choose the relationship that answers the design question. Use momentum conservation for a suitable isolated system. Use the impulse relationship when the question concerns average net force and interaction time. Use kinetic energy when the question concerns energy of motion.

Substitute values with SI units. Keep units through the calculation, round to sensible significant figures, and include direction for vector results. Finish by checking whether the result has the correct units and whether its size and direction make sense. Then explain how the result supports an improvement.

\vec{F}_{\text{net,avg}}=\frac{\Delta\vec{p}}{\Delta t}

Do not treat speed as a signed vector; velocity carries direction.
Do not claim kinetic energy is conserved in every collision.
Check the system boundary before applying conservation of momentum.

Worked example

A joining-cart bumper

Two carts collide on a low-friction track and latch together. Cart A has mass

2.0\,\mathrm{kg}

and moves right at

3.0\,\mathrm{m/s}

. Cart B has mass

1.0\,\mathrm{kg}

and is initially at rest. Find their shared velocity and compare kinetic energy before and after.

Define the system
The system is both carts. Use the track as the reference frame and choose right as positive. Assume external horizontal forces are negligible during the short collision. The unknown is the shared final velocity.
Apply momentum conservation
The carts latch together, so they have one final velocity. Momentum is conserved under the stated system assumption.
$(2.0\,\mathrm{kg})(+3.0\,\mathrm{m/s})+(1.0\,\mathrm{kg})(0\,\mathrm{m/s})=(3.0\,\mathrm{kg})v_f$
Calculate the shared velocity
Divide the initial momentum by the combined mass. The positive sign means the carts move to the right.
$v_f=+2.0\,\mathrm{m/s}$
Compare kinetic energy
The initial kinetic energy is $9.0\,\mathrm{J}$ . The final kinetic energy is $6.0\,\mathrm{J}$ . The $3.0\,\mathrm{J}$ difference is transferred to other forms, such as sound and deformation. A bumper could be assessed by whether it reduces unwanted impacts or damage; this calculation alone does not test a real bumper.
$E_{k,i}=9.0\,\mathrm{J},\qquad E_{k,f}=6.0\,\mathrm{J}$

Answer: The carts move together at

2.0\,\mathrm{m/s}

to the right. Their kinetic energy decreases by

3.0\,\mathrm{J}

.

Check: The units reduce to velocity for the momentum calculation. The final speed is between the carts’ initial speeds, and the direction is right. The energy decrease is consistent with a collision in which the carts latch together.

Worked example

Comparing two vehicle stopping distances

A vehicle of mass

1.20\times10^3\,\mathrm{kg}

moves at

15.0\,\mathrm{m/s}

and stops. Compare the average net stopping force if its stopping distance is

0.75\,\mathrm{m}

versus

1.50\,\mathrm{m}

. Use the work-energy relationship for average force over the stopping distance.

Define the system and direction
The system is the vehicle. Use the road as the reference frame and choose the direction of motion as positive. The final speed is zero. The average net force points opposite the motion.
Find the initial kinetic energy
The final kinetic energy is zero. The initial kinetic energy must be removed from the vehicle’s motion during stopping.
$E_{k,i}=\frac{1}{2}(1.20\times10^3\,\mathrm{kg})(15.0\,\mathrm{m/s})^2=1.35\times10^5\,\mathrm{J}$
Calculate the force for each distance
The magnitude of the average stopping force times stopping distance equals the kinetic energy removed. The force direction is negative in the chosen frame.
$F_{\text{avg}}=-\frac{1.35\times10^5\,\mathrm{J}}{0.75\,\mathrm{m}}=-1.8\times10^5\,\mathrm{N};\quad F_{\text{avg}}=-\frac{1.35\times10^5\,\mathrm{J}}{1.50\,\mathrm{m}}=-9.0\times10^4\,\mathrm{N}$
Interpret the comparison
Doubling the stopping distance halves the calculated average net force for the same initial energy. A design that allows more controlled stopping distance could therefore reduce average force in this model. These values are calculations, not measurements of a vehicle or safety system.

Answer: The average forces are

1.8\times10^5\,\mathrm{N}

and

9.0\times10^4\,\mathrm{N}

, both opposite the vehicle’s motion.

Check: A joule per metre is a newton, as required. The longer stopping distance gives the smaller force. The magnitudes are large, which is plausible for stopping a moving vehicle over a short distance.

Worked example

Improving a package-catching net

A

5.0\,\mathrm{kg}

package moves downward at

4.0\,\mathrm{m/s}

and is brought to rest by a net. Compare the average net force on the package if the stopping time is

0.40\,\mathrm{s}

or

0.80\,\mathrm{s}

. Take upward as positive.

Set the system and signs
The system is the package, and the reference frame is the ground. Up is positive. The initial velocity is negative and the final velocity is zero. The unknown is average net force on the package.
Calculate the momentum change
The package’s initial momentum is downward. Its final momentum is zero, so the change in momentum is upward.
$\Delta p=(5.0\,\mathrm{kg})[0-(-4.0\,\mathrm{m/s})]=+20\,\mathrm{kg\,m/s}$
Compare average net forces
Divide the same momentum change by each stopping time. A longer stopping time produces a smaller average net force.
$F_{\text{avg},1}=\frac{20\,\mathrm{kg\,m/s}}{0.40\,\mathrm{s}}=+50\,\mathrm{N};\qquad F_{\text{avg},2}=\frac{20\,\mathrm{kg\,m/s}}{0.80\,\mathrm{s}}=+25\,\mathrm{N}$
Relate the result to the design
The net force includes all forces on the package, including gravity. In this model, a net that increases stopping time reduces the average net force during the catch. A real net would need appropriate testing to confirm its performance.

Answer: The average net force is

50\,\mathrm{N}

upward for

0.40\,\mathrm{s}

and

25\,\mathrm{N}

upward for

0.80\,\mathrm{s}

.

Check: The units are newtons, and the upward direction matches the required change from downward motion to rest. Doubling the time halves the average net force for the same momentum change.

Common mistakes and how to avoid them

Treating momentum as a positive magnitude regardless of direction.

Correction: Assign a positive direction first. Use signed velocity components so momentum changes have the correct direction.

Assuming kinetic energy is conserved whenever momentum is conserved.

Correction: Momentum can be conserved while kinetic energy is transferred to sound, deformation, or thermal energy.

Saying a cushion removes the momentum change.

Correction: If an object starts and ends at the same velocities, its momentum change is unchanged. A cushion can increase the time or distance over which the change occurs.

Presenting a model calculation as a completed product test.

Correction: Label calculated or simulated results as model results. Identify actual measurements separately and do not claim testing that was not done.

Lesson summary

Define the system, frame, positive direction, and assumptions before calculating.
Use momentum conservation for a system with negligible external force during the interaction.
Use kinetic energy to estimate the energy of motion that a technology must manage.
Use impulse to relate momentum change to average net force and time.
Make a design claim that follows from the calculation, and state what still needs testing.

Check your understanding

Question 1

A ball’s momentum changes by

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

over

0.30\,\mathrm{s}

. What is the magnitude of its average net force?

$4.0\,\mathrm{N}$
$40\,\mathrm{N}$
$3.6\,\mathrm{N}$
$36\,\mathrm{N}$

Show answer and explanation

40\,\mathrm{N}

Divide momentum change by time:

12\,\mathrm{kg\,m/s}\div0.30\,\mathrm{s}=40\,\mathrm{N}

. The direction would depend on the direction of the momentum change.

Question 2

Two objects latch together in a collision. Which statement is generally correct for a suitable isolated system?

Momentum is conserved, and kinetic energy must be conserved.
Kinetic energy is conserved, but momentum must decrease.
Momentum is conserved, while some kinetic energy may transfer to other forms.
Both momentum and total energy disappear during the collision.

Show answer and explanation

Momentum is conserved, while some kinetic energy may transfer to other forms.

Momentum is conserved when external forces are negligible for the system. In a collision where objects latch together, kinetic energy can transfer to other forms.

Key terms

System: The object or group of objects selected for analysis.
Reference frame: The viewpoint used to describe position and motion.
Momentum: A vector quantity equal to an object’s mass multiplied by its velocity.
Kinetic energy: The scalar energy an object has because it is moving.
Impulse: The change in momentum of an object or system.
Average net force: The average combined force on an object over a stated time interval.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C1.1. 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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C1.1 · Analyse and improve a technology using energy and momentum

Ontario Grade 12 Physics

A Grade 12 approach to analysing how designs transfer energy and change motion

What you will learn

1. From motion ideas to a useful model

2. Conservation and technology choices

3. Turning calculations into design improvements

4. A reliable analysis routine

A joining-cart bumper

Comparing two vehicle stopping distances

Improving a package-catching net

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH4U

About this lesson and its review