Relative velocity in one dimension
Subtract signed velocities in the requested observer frame.
A faster cyclist passes a slower cyclist.
Need an earlier step? Start with reference frames →
Relative to the road, Maya rides right at 5 m/s and Leo rides right at 3 m/s. In 1 second Maya gains 2 m on Leo. Leo describes Maya’s velocity as +2 m/s relative to himself.
Words and symbols you will use
- v_object
- Object velocity measured in the ground frame.
- v_observer
- Observer velocity measured in that same ground frame.
- v_relative
- Object velocity seen by the observer: v_object − v_observer.
Read the picture, one step at a time.
- Use the same positive direction and the same ground frame for both velocities.
- In 1 s Maya advances 5 m; Leo advances 3 m.
- The separation grows by 5 − 3 = 2 m in 1 s. Maya’s velocity relative to Leo is +2 m/s.
| Compare | What it means | What follows |
|---|---|---|
| Maya relative to road | Ground description | +5 m/s |
| Maya relative to Leo | 5 − 3 | +2 m/s |
| Leo relative to Maya | 3 − 5 | −2 m/s |
The idea to keep: Always subtract signed velocities, not just speeds. If the observer moves left, their negative velocity changes the subtraction. This lesson keeps relative-velocity calculations one-dimensional.
How do I calculate velocity relative to a moving observer?
Subtract the observer’s ground-frame velocity from the object’s ground-frame velocity, using the same positive direction for both.
Subscripts help prevent sign mistakes: v_A/B means the velocity of A measured by B. If both velocities are known relative to ground, v_A/B = v_A/G − v_B/G. A negative result means A moves in the negative direction in B’s frame.
Objects moving in the same direction can still move relative to each other. A faster runner gains on a slower runner. If their directions oppose, their separation can change at the sum of the speed magnitudes. The sign still follows the coordinate convention.
A reliable approach
- Name which velocity is requested: object relative to which observer?
- Express both known velocities in one shared ground frame.
- Subtract in the correct order, then interpret the sign.
Work through one example.
Car A moves east at 20 m/s and car B east at 12 m/s. Find A relative to B, then B relative to A.
Follow the worked solution
- With east positive, v_A/B = 20 − 12 = +8 m/s.
- v_B/A = 12 − 20 = −8 m/s.
- The relative velocities have equal magnitudes and opposite directions.
Subtracting speed magnitudes without assigning directions fails for objects moving in opposite directions. Relative-velocity calculations in this unit are restricted to one dimension.
Explain it without notes: Reverse object and observer and explain what happens to the result.
Try explaining it now.
A cyclist has ground velocity +9 m/s and a runner −2 m/s. Is cyclist relative to runner +7 m/s?
Compare with an explanation
No. Subtract the signed observer velocity: 9 − (−2) = +11 m/s.
Another explanation, if you need oneOptional external lesson · Flipping Physics
Relative motion refresher
This lesson is complete without a video. For another teacher’s explanation, open the original resource below. Pause after a diagram and explain the idea in your own words.
Suggested section 20:13–23:54. Video by Flipping Physics / Jonathan Thomas-Palmer. Refresh Kid is not affiliated with or endorsed by Flipping Physics. These links open another website. Some videos use g = 9.81 m/s²; our examples state g = 10 m/s². Follow the value given in each problem.
After watching: Reverse object and observer and explain what happens to the result.
Make a prediction. Test it.
Set object ground velocity +8 m/s and observer ground velocity +10 m/s. Predict the relative sign before switching observers.
Motion graphs · same clock, different quantities
Try two questions.
Two original Refresh Kid questions. Choose an answer, explain it to yourself, then check the reasoning. These are not released AP exam questions.
Show your reasoning.
Original mini-FRQ · 4 points · Self-checkA train travels at +10 m/s relative to ground. A passenger walks at −2 m/s relative to the train.
- Find passenger velocity relative to ground.
- Find train velocity relative to passenger.
- State the passenger’s ground-frame direction.
- Explain why walking backward in the train need not mean moving backward relative to ground.
Your response stays on this page and is not submitted or automatically graded. Copy it before leaving.
Compare with the worked solution & scoring guide
- 1 point: v_P/G = −2 + 10 = +8 m/s.
- 1 point: v_T/P = +2 m/s.
- 1 point: The passenger moves in the positive direction relative to ground.
- 1 point: The train’s forward velocity exceeds the passenger’s backward relative velocity.
Accept an equivalent correct method. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Does subtraction order matter?
Yes. Swapping object and observer reverses the relative velocity.
RECALL 2When is relative velocity zero?
When object and observer have equal velocities in the common frame.
RECALL 3What happens to acceleration between constant-velocity frames?
It is unchanged in this nonrelativistic model.
Come back tomorrow: answer these with the cards closed. Try again a week later, especially the ones you missed.
Keep the key ideas handy.
Relative velocity in one dimension
Core idea: Subtract the observer’s ground-frame velocity from the object’s ground-frame velocity, using the same positive direction for both.
- v_A/B = v_A/G − v_B/G
- v_A/G = v_A/B + v_B/G
Avoid this: Subtracting speed magnitudes without assigning directions fails for objects moving in opposite directions. Relative-velocity calculations in this unit are restricted to one dimension.
Remember why: Always subtract signed velocities, not just speeds. If the observer moves left, their negative velocity changes the subtraction. This lesson keeps relative-velocity calculations one-dimensional.
Both input velocities must use the same frame and direction convention.
Explain, don’t just substituteReverse object and observer and explain what happens to the result.
Refresh Kid · AP Physics 1 · Unit 1 · 1.4.B · Check units, direction and model assumptions.
Connect to released AP practice.
2026 · Question 1 · Version J
Use Part A(i) for component velocity graphs and Part A(ii) for a kinematics derivation. The remaining parts use fluid concepts from Unit 8; they are not Unit 1-only practice.
This is a Unit 1 synthesis task to revisit after learning projectile motion. Historical papers may use different timing or course coverage. Official questions remain on College Board’s site; our questions below are original practice.
Browse released years and scoring information ↗Framework alignment: College Board CED, Topic 1.4. Current exam corrections. Checked September 16, 2026. These explanations and practice items are independently authored by Refresh Kid. The simulation is a mathematical model, not experimental data.
About the videos and learning approach
Optional videos are linked to the publisher’s website and YouTube channel with credit. No external video player is loaded on this lesson page. The written lessons, simulations and practice here are independently authored by Refresh Kid.
We combine worked examples, visual models, explanation and recall practice. See the Institute of Education Sciences study guide ↗ for the underlying learning recommendations.
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