Which bicycle gear goes farther?

Change one sprocket in a bicycle model. Keep the wheel and pedalling period fixed, then compare distance travelled over model time.

Starting notebook…

Simulation · Mechanical Systems

How does front-to-rear sprocket size affect distance travelled when wheel size and pedalling period stay the same?

A bicycle chain carries motion from the front sprocket to the rear sprocket. While the drive is engaged, the rear wheel turns with the rear sprocket. The output/input speed multiplier equals front sprocket radius ÷ rear sprocket radius. For sprockets with the same chain pitch, it also equals front teeth ÷ rear teeth. This GeoGebra model controls radii, not tooth counts. The chain-linked sprockets turn in the same direction. A larger multiplier gives more wheel turns per pedal turn, but less ideal wheel torque for the same crank torque. It does not guarantee that a real rider can maintain the same pedalling rate on a hill.

Review the related lesson →

Science words you will use

A hypothesis is a possible explanation. A prediction is what you expect to observe. The independent variable is what you change or compare; the dependent variable is what you measure. Controlled variables stay the same in a fair test. Qualitative observations describe what you notice; quantitative observations use numbers and units.

Accuracy means closeness to a trusted value. Precision describes how closely repeated readings agree. Repeating a result does not prove it is accurate. A pattern between two quantities does not, by itself, show that one caused the other.

What good work includes

Record all four slider settings and change one variable at a time.
Use actual displayed model times and distances to support the sprocket-ratio comparison.
Explain the speed–turning-effect trade-off and identify what this ideal model cannot tell you about real riding.

Plan your investigation

My scientific notebook

Plan your investigation

Use your own words. You can return and change your ideas as you collect evidence.

My research questionWhat can you investigate using this equipment or record?My hypothesisGive a possible explanation: I think… because…My predictionIf… then I predict… because…What I change or compareName the independent variable. In an observation study, name what you compare.What I measureName the dependent variable and its units.What stays the sameKeep rb, rw and pr fixed while changing only rf. Reset time to the same starting point and use the actual model readouts. Record every setup so comparisons can be repeated.My procedure and changesUse repeatable steps. Record changes to the plan and why you made them.Reading my equipmentWhat is the smallest marked division? What units and rounding will you use?

Your apparatus

Test a bicycle chain drive

Use Austin Harvey’s GeoGebra Bicycle Gears model. rf is front sprocket radius; rb is rear sprocket radius; rw is wheel radius; pr is pedal rotation time. Use Start Bike and Stop Bike. Record the displayed model time and Distance Traveled in inches. These are model values, not real riding measurements. Keep rw and pr fixed while comparing sprockets.

Open GeoGebra in a new tab

On a phone, turn sideways and open the model in a new tab for larger controls. Your notebook stays on this page.

Use Start Bike / Stop Bike and the model time control. Check all four slider values after restarting. Open the model separately if phone controls are too small.

Bicycle Gears · Austin Harvey · GeoGebra. Embedded original model.

GearLab climbing extension →Gear Generator design extension →

Materials & safety

GeoGebra Bicycle Gears model
Digital notebook and calculator
GearLab cycling calculator for optional extension

This is a computer investigation. No actual cycling is required. Any later riding or equipment activity needs a separate teacher-approved plan. Never touch moving chains or gears.

Suggested procedure — adapt and record changes

Write a prediction before testing. Record rf, rb, rw and pr from the sliders. Compute the starting output/input multiplier rf ÷ rb. Use the radius values actually shown; do not describe them as tooth counts.
Keep rw and pr fixed. Start with rf = rb if the sliders allow it. Reset model time to zero and check that Distance Traveled reads zero. Use the model time control or animation buttons; do not use a wall clock.
Start the model, then pause at several points across one or more pedal rotations. Record actual displayed model times and distances in inches. If you pause late, record the actual values instead of the target time. Record the slider values in the observation field.
Reset time and change only rf to another permitted value. Keep rb, rw and pr unchanged. Use a new group label such as Front larger. Repeat at similar model times. Complete a third permitted front-radius setting.
Compare distance at matching times. If times differ, calculate distance ÷ model time as a comparison only when motion is steady; do not divide the zero-time row. Identify which setup gives more wheel travel per pedal rotation. Check the direction of each sprocket.
Revisit one setup to check your reading. This is a repeat of a deterministic model, not new evidence from real riding. Then compare different radius pairs with the same rf ÷ rb ratio while keeping rw and pr fixed.
Explain the gearing trade-off. A larger distance at fixed pedalling rate is not proof of easier climbing. Changing wheel radius or pedal period is a separate investigation; keep those changes out of the first comparison. Label conclusions as predictions of an ideal model.

Collect · check · compare

Your data and graph

Record the model’s displayed time and distance, including valid zeros. Time uses the applet’s displayed model scale; do not assume it is real riding time. Keep a separate label for each sprocket setup. Do not invent readings if a label cannot be read.

Graph titleGraph view

The table keeps your exact readings. On a small screen, scroll inside the table or graph to see the rest. With a keyboard, focus it and use the arrow keys.

Add readings to enable data and graph downloads.

Explain your results and reflect

Think like a scientist

Explain, question and reflect

Patterns in my resultsCompare at least three front-radius settings at matching model times. Show rf ÷ rb and use distance readings to support your conclusion. Explain the effect of wheel size separately from the sprocket multiplier.Limits and possible errorsDiscuss reading and pausing limits, and the effects missing from the model: rider effort, hills, wind, friction and tyre grip. Replaying this model is not an independent trial on a bicycle.My conclusionAnswer your research question. Your result does not have to match your prediction.The evidenceUse measurements, units and trial numbers to support your conclusion.Why the evidence supports my conclusionExplain how chain motion links front sprocket, rear sprocket and wheel. Connect radius ratio to wheel turns. Separate what the model shows from claims about how hard a real rider must pedal.What I would improve or test nextWhat would you change? Why? Revisit your hypothesis and prediction.

A new question to explore

Use GearLab.bike as an advanced climbing scenario. Keep rider, wheel, power, wind and gradient fixed when comparing drivetrain options. Its predicted road speed depends on those assumptions. Gear Generator is an optional meshing-gear design extension, not the bicycle chain model.

Keep your work

Your lab report

Preview your notes, measurements, calculations and graph. Choose Print / Save as PDF in the report, or download an HTML copy. Submit that PDF in Google Classroom using your teacher’s instructions. This website does not submit your work.

Student name (optional)Class (optional)Investigation date (optional)

Restore notebook backup

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