2. Computer simulations

What Is a Computer Simulation?

A computer simulation is a digital model used to represent the behaviour of a real or imagined physical system.

The computer uses:

• Mathematical equations.
• Scientific laws.
• Starting conditions.
• Assumptions.
• Input values.

to calculate how a system is expected to behave.

For example, a simulation could model:

• A ball falling under gravity.
• A projectile moving through the air.
• Planets orbiting a star.
• Electric circuits.
• Collisions between objects.
• Waves moving through a medium.
• Heat transfer.
• Gas particles moving inside a container.

Instead of physically performing every experiment, scientists can sometimes investigate the system using a computer model.

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Models and Simulations

A model is a simplified representation of a system.

A simulation uses a model to investigate how that system behaves.

For example, a model of projectile motion might use:

horizontal velocity

vertical velocity

gravitational acceleration

time

The computer repeatedly calculates the projectile's position as time passes.

The resulting simulation can show the object's trajectory.

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Simulations Are Based on Rules

A simulation does not simply "guess" what will happen.

It follows rules provided by the model.

For example, a basic falling-object simulation might use:

g = 9.8 m/s²

and:

v = u + at

The computer calculates new values as time passes.

A more advanced model might also include:

• Air resistance.
• Wind.
• Changing gravitational field strength.
• Object shape.
• Air density.

The quality of the simulation depends strongly on the quality of the model.

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Inputs, Processing, and Outputs

A useful way to understand simulations is:

Inputs → Model → Calculations → Outputs

Inputs

Values supplied to the simulation.

For example:

• Mass.
• Initial velocity.
• Angle.
• Temperature.
• Force.
• Resistance.

Model

The equations and assumptions describing the system.

Calculations

The computer uses the model to calculate what happens.

Outputs

The results may appear as:

• Numbers.
• Tables.
• Graphs.
• Animations.
• Diagrams.
• Predicted values.

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Initial Conditions

The initial conditions describe the state of the system when the simulation begins.

For projectile motion, these might include:

initial speed = 20 m/s

launch angle = 40°

initial height = 2 m

Changing the initial conditions can change the outcome of the simulation.

This allows scientists to investigate many different situations quickly.

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Changing One Variable

Simulations are particularly useful for investigating how one variable affects another.

For example:

How does launch angle affect projectile range?

A student could keep constant:

• Initial speed.
• Initial height.
• Gravity.

Then change only:

launch angle

Possible values:

10°, 20°, 30°, 40°, 50°, 60°

The simulation could calculate the range for each angle.

This creates a controlled virtual investigation.

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Example: Projectile Motion

Suppose a simulation launches a ball at:

20 m/s

Students can vary the launch angle and observe:

• Maximum height.
• Time of flight.
• Horizontal range.
• Shape of the trajectory.

They can collect results in a table:

Students can then graph:

launch angle against range

The simulation becomes a source of experimental-style data.

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Simulations Can Make the Invisible Visible

Some physics concepts are difficult to observe directly.

Examples include:

• Electric fields.
• Magnetic fields.
• Atomic particles.
• Molecular motion.
• Energy transfers.
• Forces.
• Wave interference.

Simulations can represent these using visual models.

For example, an electric-field simulation might display arrows showing the direction and relative strength of the field around charges.

The arrows are not physically visible in reality. They are a representation created by the model.

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Simulations and Particle Models

Imagine investigating the behaviour of gas particles.

A simulation might represent particles as moving circles.

Students could change:

• Temperature.
• Volume.
• Number of particles.

The simulation might show changes in:

• Particle speed.
• Collision frequency.
• Pressure.

This can help connect the particle model to observable properties of gases.

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Simulating Motion

Motion simulations can display:

• Position.
• Velocity.
• Acceleration.
• Forces.
• Time.

Students can sometimes pause the simulation and examine individual moments.

This is useful because real motion may happen too quickly to observe carefully.

For example, a collision lasting:

0.05 s

may be difficult to study directly.

A simulation can display the collision slowly or one step at a time.

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Simulating Forces

A force simulation might allow students to change:

• Applied force.
• Mass.
• Friction.
• Surface.
• Direction of force.

Students could investigate Newton's second law:

F = ma

For example, keep mass constant and increase resultant force.

The model should predict that acceleration increases.

Then keep force constant and increase mass.

The model should predict that acceleration decreases.

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Simulating Circuits

Circuit simulations can allow students to construct virtual circuits using:

• Cells.
• Batteries.
• Resistors.
• Lamps.
• Switches.
• Ammeters.
• Voltmeters.

Students can investigate relationships such as:

V = IR

without needing to physically rebuild a circuit after every change.

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Simulations and Real Experiments

Computer simulations and physical experiments are not the same thing.

A physical experiment investigates the actual behaviour of a real system.

A simulation investigates the behaviour predicted by a model.

This distinction is extremely important.

A simulation result tells us:

what the model predicts

Experimental evidence tells us:

what the physical system actually did

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Comparing Simulation and Experimental Data

Scientists often compare simulation predictions with measurements from real experiments.

Suppose a simulation predicts that a cart will accelerate at:

2.5 m/s²

A physical experiment gives:

2.3 m/s²

The results are similar but not identical.

Scientists might investigate the difference.

Possible explanations include:

• Friction.
• Air resistance.
• Measurement uncertainty.
• Simplifications in the model.
• Sensor errors.
• Uneven surfaces.

Differences between simulation and experiment can help scientists improve their models.

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A Simple Comparison

Imagine a falling-object simulation that ignores air resistance.

The model predicts:

a = 9.8 m/s²

A real falling object may behave differently because air resistance acts on it.

For a dense metal ball falling a short distance, the difference might be small.

For a sheet of paper, the difference could be very large.

The simulation is not necessarily "wrong." It may simply be based on assumptions that do not adequately represent that situation.

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Assumptions

An assumption is something accepted as true within the model to simplify the situation.

Common physics assumptions include:

• No air resistance.
• No friction.
• Constant gravitational field strength.
• Perfectly rigid objects.
• Perfectly elastic collisions.
• Point masses.
• Ideal gases.
• Ideal electrical components.

Assumptions make models easier to use but may also reduce their realism.

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Ideal Models

Physics often begins with an idealized model.

For example, a projectile model might assume:

air resistance = 0

This makes the mathematics much simpler.

The model can still be extremely useful for understanding the basic physics.

Later, a more sophisticated model could include drag.

This illustrates an important principle:

A useful model does not need to include every detail of reality.

It needs to include enough detail for its intended purpose.

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Model Complexity

Adding more variables can make a simulation more realistic.

For a falling object, a simple model might include:

• Gravity.

A more complex model might include:

• Gravity.
• Air resistance.
• Air density.
• Wind.
• Object shape.
• Changing atmospheric density.

However, increased complexity also creates challenges.

More complex simulations may require:

• More information.
• More measurements.
• More computing power.
• More assumptions.
• More difficult interpretation.

The most complex model is not automatically the best model.

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Simulation Results as Data

Simulation results should be treated as data that need interpretation.

For example:

Students can:

• Identify patterns.
• Calculate relationships.
• Construct graphs.
• Make predictions.
• Compare results with equations.
• Compare results with experimental measurements.

The fact that data were generated by a computer does not remove the need for scientific analysis.

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Interpreting Simulation Graphs

Suppose a simulation generates a velocity-time graph.

A student might determine:

• Initial velocity.
• Final velocity.
• Acceleration.
• Time interval.
• Displacement from the area under the graph.

Simulation graphs should be interpreted using the same physics principles as graphs produced from real experimental data.

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Testing Predictions

A simulation can be used to make a prediction before conducting an experiment.

For example:

1. Create a model of a pendulum.
2. Predict its period.
3. Perform a real pendulum experiment.
4. Measure the actual period.
5. Compare the results.

This allows students to evaluate how well the model represents reality.

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Validation

Validation involves checking whether a model produces results that agree sufficiently well with real-world observations or experimental data for its intended purpose.

Suppose a model predicts:

temperature after 5 min = 65°C

The experiment gives:

64°C

That may suggest the model represents the system reasonably well under those conditions.

However, agreement in one situation does not prove that the model will work under every possible condition.

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Improving a Model

Suppose a projectile simulation consistently predicts ranges greater than those measured experimentally.

Students might hypothesize that the model is missing:

air resistance

The model could then be modified to include drag.

New predictions could be compared with the experimental data.

This process can continue:

Model → Prediction → Experiment → Comparison → Improvement

This cycle is an important part of scientific modelling.

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Strengths of Computer Simulations

Safety

Simulations can investigate situations that would be dangerous to reproduce.

Examples include:

• High-speed collisions.
• Extreme temperatures.
• High voltages.
• Spacecraft trajectories.
• Nuclear processes.

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Speed

A process that takes hours, years, or millions of years can sometimes be represented much more quickly.

For example, simulations can investigate:

• Planetary motion.
• Climate processes.
• Stellar evolution.
• Long-term orbital changes.

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Repeatability

The same simulation can be repeated using identical starting conditions.

This makes comparisons easier.

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Variable Control

One variable can often be changed while others remain constant.

This is useful for investigating cause-and-effect relationships.

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Large Numbers of Trials

Computers can perform many simulations rapidly.

Instead of testing five conditions, scientists might test:

hundreds or thousands

depending on the model.

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Investigation of Extreme Conditions

Some situations cannot easily be created in a school laboratory.

A simulation might investigate:

• Motion on the Moon.
• Very high speeds.
• Extreme pressures.
• Large gravitational fields.
• Very small particles.

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Visualization

Simulations can represent quantities that cannot be directly seen.

Examples include:

• Fields.
• Forces.
• Particle motion.
• Energy.
• Wavefronts.

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Limitations of Computer Simulations

Simulations Depend on the Model

A simulation can only behave according to the rules programmed into it.

If the model is poor, the results may also be poor.

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Simplifying Assumptions

Real systems may contain effects that the simulation ignores.

For example:

simulation → no friction

real experiment → friction exists

This can produce differences.

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Input Data May Be Incorrect

A sophisticated model can still produce poor predictions if its input values are inaccurate.

This idea is sometimes summarized as:

poor input → poor output

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Simulations May Not Include Unknown Factors

Scientists cannot deliberately include a physical effect that they do not yet know exists.

Real experiments can sometimes reveal unexpected behaviour that was not included in the model.

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Numerical Approximations

Complex simulations often solve equations approximately rather than exactly.

The computer may calculate the system in many small time steps.

Smaller steps can sometimes improve accuracy but require more computation.

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Computing Limitations

Very detailed simulations may require:

• Powerful computers.
• Large amounts of memory.
• Long processing times.

Scientists often need to balance realism against computational cost.

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Simulations Do Not Replace Experiments

A simulation is most powerful when combined with experimental evidence.

Physical experiments allow scientists to test whether a model actually describes nature.

Simulations allow scientists to explore the consequences of the model.

Together they provide complementary evidence.

A useful relationship is:

Theory → Model → Simulation → Prediction → Experiment → Comparison

The results can then be used to improve the theory or model.

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Simulations in Experimental Design

Scientists can use simulations before performing an experiment.

A simulation may help determine:

• Useful ranges of variables.
• Appropriate equipment.
• Expected results.
• Possible safety concerns.
• How many measurements may be needed.

For example, if a simulation predicts that a cart will reach approximately:

15 m/s

scientists can ensure that their motion sensor is capable of measuring that speed.

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Simulations Can Save Resources

Physical experiments may require:

• Materials.
• Equipment.
• Energy.
• Time.
• Laboratory space.

A simulation can allow many preliminary investigations without consuming these resources.

Researchers can then select the most useful conditions for physical testing.

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Simulation Example: Pendulum

Suppose students investigate:

How does pendulum length affect period?

A simulation could allow them to change:

length

while keeping constant:

• Mass.
• Starting angle.
• Gravity.

Students record the period at several lengths.

They can then perform a real pendulum experiment and compare the results.

If the results differ, students should investigate possible reasons.

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Simulation Example: Springs

A spring simulation could allow students to change:

• Spring constant.
• Applied force.
• Mass.
• Extension.

Students could test:

F = kx

A graph of force against extension should be linear for an ideal spring obeying Hooke's law.

A real spring may behave differently near or beyond its elastic limit.

This difference reveals a limitation of the ideal model.

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Simulation Example: Electric Fields

Electric fields cannot be directly seen.

A simulation can represent them using:

• Field lines.
• Arrows.
• Colour maps.
• Test charges.

Students can move charges and observe how the model predicts the field will change.

This can make an abstract concept easier to investigate.

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Simulation Example: Waves

A wave simulation might allow students to change:

• Frequency.
• Amplitude.
• Wavelength.
• Wave speed.
• Tension.

Students can observe:

• Reflection.
• Interference.
• Standing waves.
• Wave propagation.

The simulation allows variables to be changed rapidly while displaying the predicted result immediately.

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PhET Simulations

One widely used source of interactive science simulations is PhET Interactive Simulations from the University of Colorado Boulder.

PhET includes simulations covering topics such as:

• Motion.
• Forces.
• Energy.
• Electricity.
• Waves.
• Sound.
• Gravity.
• Atomic physics.

These can be useful for both teacher demonstrations and student investigations.

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Designing a Simulation Investigation

Suppose you are investigating:

How does mass affect acceleration when resultant force remains constant?

Question

How does increasing mass affect acceleration?

Independent Variable

Mass.

Dependent Variable

Acceleration.

Controlled Variable

Resultant force.

Method

1. Select a constant force.
2. Choose an initial mass.
3. Run the simulation.
4. Record acceleration.
5. Increase mass.
6. Repeat.
7. Create a table.
8. Plot acceleration against mass.
9. Identify the relationship.

Students could then compare their simulation results with:

a = F/m

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Comparing Simulation and Experimental Results

Suppose the results are:

The simulation consistently predicts slightly greater acceleration.

A useful scientific question is:

Why?

Possible explanations might include:

• Friction in the real experiment.
• Air resistance.
• Measurement uncertainty.
• An imperfect track.
• Simplifications in the simulation.

The disagreement itself provides useful information.

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Percentage Difference

Simulation and experimental results can sometimes be compared quantitatively.

For example:

simulation = 5.0 m/s²

experiment = 4.7 m/s²

Difference:

5.0 − 4.7 = 0.3 m/s²

The percentage difference relative to the simulation result is:

(0.3/5.0) × 100

= 6%

This gives a numerical measure of how different the results are.

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Sensitivity

A model may be sensitive to certain inputs.

This means that a small change in an input produces a large change in the output.

For example, suppose a simulation produces:

input = 5.0 → output = 20

but:

input = 5.1 → output = 35

The model is highly sensitive to that input in this region.

Scientists often investigate sensitivity to determine which measurements must be especially accurate.

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Uncertainty in Simulations

Simulation outputs may appear extremely precise.

For example:

18.437295 m

But the model might use uncertain input data.

If the starting measurement was only known to approximately:

18 ± 1 m

reporting many decimal places in the output can create a false impression of certainty.

Computer-generated precision is not necessarily physical accuracy.

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Interpreting Animations Carefully

An animation can make a simulation appear realistic.

However:

realistic appearance ≠ accurate scientific model

The scientific value of the simulation depends on:

• Equations.
• Assumptions.
• Input data.
• Validation.

Graphics alone do not determine whether a model is reliable.

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Simulations and Prediction

Simulations are often used to answer questions of the form:

"What does our model predict will happen if...?"

For example:

• What happens if mass doubles?
• What happens if friction decreases?
• What happens if voltage increases?
• What happens if gravity is weaker?
• What happens if the launch angle changes?

The simulation allows systematic exploration of these questions.

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Common Mistakes

Treating a Simulation as Reality

A simulation represents a model of reality, not reality itself.

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Assuming Computer Results Must Be Correct

Computers accurately follow their instructions.

If the model, assumptions, input data, or programming are incorrect, the result can also be incorrect.

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Ignoring Assumptions

Always ask:

What does this model assume?

For example:

• Is friction ignored?
• Is air resistance included?
• Is gravity constant?

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Changing Several Variables at Once

If you want to investigate the effect of one variable, keep other important variables constant.

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Looking Only at the Animation

Collect numerical data whenever possible.

Use:

• Tables.
• Graphs.
• Measurements.
• Calculations.

Do not rely only on what the animation appears to show.

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Ignoring Experimental Evidence

Simulation predictions should be compared with real measurements whenever practical.

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Assuming More Complex Means Better

A more complicated model may not be more useful if the additional complexity does not improve the prediction needed.

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Confusing Precision with Accuracy

A computer may display many decimal places.

That does not mean the prediction is equally accurate.

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Check Your Understanding

1. What is a computer simulation?

2. Explain the difference between a model and a simulation.

3. Give four examples of physical systems that could be investigated using simulations.

4. What are initial conditions?

5. Identify the inputs and possible outputs of a projectile-motion simulation.

6. Explain how a simulation could be used to investigate Newton's second law.

7. Why are simulations useful for studying electric fields?

8. Explain how simulations can make invisible processes easier to understand.

9. What is an assumption in a scientific model?

10. Give two assumptions that might be used in a projectile-motion simulation.

11. Explain why a simulation ignoring air resistance may still be useful.

12. Give three advantages of computer simulations.

13. Give three limitations of computer simulations.

14. Explain why simulation results should be compared with experimental data.

15. A simulation predicts an acceleration of 4.0 m/s² while an experiment gives 3.6 m/s². Suggest two possible reasons for the difference.

16. Explain what it means to validate a model.

17. Why does agreement between a model and one experiment not prove that the model is correct in every situation?

18. Explain how simulations can improve experimental design.

19. Why can simulations be useful for dangerous experiments?

20. Explain how a simulation could save time or resources.

21. What is meant by sensitivity in a computer model?

22. Why should a simulation result containing many decimal places not automatically be considered highly accurate?

23. A simulation of a spring perfectly follows F = kx at all extensions, but a real spring stops following this relationship at large extensions. Explain why.

24. Design a simulation investigation into one physics relationship. Identify the independent, dependent, and controlled variables.

25. Explain why simulations and physical experiments are often more useful when used together.

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Key Terms

• Computer simulation – computer-based representation of the behaviour of a system.
• Model – simplified representation of a real system or process.
• Input – information or values supplied to a model.
• Output – results generated by a model or simulation.
• Initial conditions – state of a system at the beginning of a simulation.
• Assumption – condition accepted within a model to simplify the system.
• Idealized model – simplified model that intentionally ignores some real-world effects.
• Validation – comparison of model predictions with observations or experimental evidence.
• Prediction – expected result produced using a model or scientific theory.
• Sensitivity – degree to which changes in inputs affect model outputs.
• Simulation data – values generated by running a computer model.
• Experimental data – measurements collected from a physical investigation.
• Numerical model – model in which mathematical equations are solved computationally.
• Variable – quantity that can change within an investigation or model.

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Key Takeaways

• Computer simulations use mathematical models to represent physical systems.
• Simulations take inputs, perform calculations using a model, and generate outputs.
• Initial conditions can strongly affect simulation results.
• Simulations can investigate motion, forces, circuits, waves, fields, particles, heat, and many other systems.
• Simulations allow variables to be changed systematically.
• They can represent processes that are too fast, slow, large, small, expensive, or dangerous to investigate directly.
• Simulations can make invisible quantities easier to visualize.
• Simulation results are predictions of a model, not direct observations of reality.
• Models usually contain simplifying assumptions.
• Comparing simulation results with experimental data helps scientists evaluate models.
• Differences between predictions and measurements can reveal missing factors or weaknesses in a model.
• Models can be improved using experimental evidence.
• Simulations can help scientists design experiments before conducting them.
• More complicated models are not automatically better models.
• Computer-generated precision does not guarantee physical accuracy.
• Simulation data should be analyzed using tables, graphs, calculations, and scientific reasoning.
• Physical experiments remain essential for testing whether models accurately represent nature.
• Simulations and experiments are complementary scientific tools.

A useful way to think about computer simulation is:

The simulation tells us what the model predicts. The experiment tells us what nature actually does. Comparing the two helps us improve our understanding.