Choosing constant-acceleration equations
Choose an equation from the model and known quantities.
Give the cart the same velocity change every second.
Need an earlier step? Start with velocity & acceleration graphs →
A cart starts from rest. Its acceleration stays +2 m/s², so its velocity is +2 m/s after 1 s and +4 m/s after 2 s. During the 2 s, it travels 4 m—not 8 m—because it was not moving at its final velocity the whole time.
Words and symbols you will use
- v₀ / v
- Initial / final velocity, in m/s.
- a and t
- Constant acceleration (m/s²) and elapsed time (s).
- Δx
- Final position minus initial position, in meters.
Read the picture, one step at a time.
- Velocity rises along a straight line from 0 to 4 m/s in 2 s.
- The average velocity is halfway between the endpoints: 2 m/s, because acceleration is constant.
- Displacement is the triangular area: ½ × 2 s × 4 m/s = 4 m.
| Compare | What it means | What follows |
|---|---|---|
| v = v₀ + at | No displacement needed | Find how velocity changes. |
| Δx = v₀t + ½at² | No final velocity needed | Find position change over time. |
| v² = v₀² + 2aΔx | No time needed | Connect velocity and displacement. |
The idea to keep: The equations summarize the same motion. Check that acceleration stays constant, choose a positive direction, then list knowns and the unknown before substituting.
Which kinematics equation should I use?
First verify constant acceleration. Then choose the equation containing your known quantities and the unknown you need, with the fewest extra unknowns.
Equations are models with conditions, not a menu to use blindly. A braking car can be approximated with constant acceleration during a specified interval. If the braking changes, the same formula may not model the entire motion.
Choose the axis before substituting and carry signed components throughout. Displacement can be negative; elapsed time must fit the described interval. An equation containing v² can lose directional information, so decide the sign of velocity from the physical situation.
Check dimensions before calculating: (m/s)·s and (m/s²)·s² both give metres. Convert every input to compatible units first; 54 km/h × (1000 m/km) × (1 h/3600 s) = 15 m/s. Keep extra digits during working and round the final answer sensibly.
A reliable approach
- List x₀, x, v₀, v, a and elapsed time with units.
- Check that one constant a applies over the interval.
- Solve symbolically first, then substitute and check direction, units and plausible size.
Work through one example.
A cart moves at +3 m/s, accelerates constantly at +2 m/s², and travels +12 m. Find its final velocity without finding time first.
Follow the worked solution
- Use v² = v₀² + 2aΔx.
- v² = 9 + 2(2)(12) = 57 m²/s².
- v = +√57 ≈ +7.55 m/s; the positive root fits continued positive motion.
v_avg = (v₀ + v)/2 is valid for constant acceleration, not arbitrary motion.
Explain it without notes: Explain why a correct equation can still give an inappropriate physical answer.
Try explaining it now.
You know initial velocity, acceleration and displacement, but not time. Must you find time first?
Compare with an explanation
No. For constant acceleration, v² = v₀² + 2aΔx omits time. Choose the final velocity sign using the physical direction.
Another explanation, if you need oneOptional external lesson · Flipping Physics
Recognizing the motion model
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 14:31–16:57. 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: Explain why a correct equation can still give an inappropriate physical answer.
Make a prediction. Test it.
Set v₀ = 8 m/s and a = −2 m/s². Inspect t = 4 s, then t = 5 s. Distinguish the mathematical continuation from a braking model that ends at rest.
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-checkAn ideal position tracker gives x = 1.0, 2.0, 5.0, 10.0 m at t = 0, 1, 2, 3 s.
- Test whether x = x₀ + ½at² with release from rest fits.
- Choose transformed axes that would give a straight line.
- Relate that line’s slope to a.
- Explain one reason real measurements might depart from the line.
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: x₀ = 1 m and a = 2 m/s² reproduce all four points. This is a compatible model, not proof of unobserved motion.
- 1 point: Plot x − x₀ against t²: the pairs are (0,0), (1,1), (4,4), (9,9) in s² and m.
- 1 point: Slope = 1 m/s² = a/2, so a = 2 m/s².
- 1 point: Nonzero release velocity, nonconstant acceleration, distance calibration or timing uncertainty can cause departures; identify which would affect the measurement/model.
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 1Which equation omits time?
v² = v₀² + 2aΔx.
RECALL 2What must stay constant?
Acceleration over the modeled interval.
RECALL 3How do you select a square-root sign?
Use the direction implied by the motion, not an automatic positive root.
Come back tomorrow: answer these with the cards closed. Try again a week later, especially the ones you missed.
Keep the key ideas handy.
Choosing constant-acceleration equations
Core idea: First verify constant acceleration. Then choose the equation containing your known quantities and the unknown you need, with the fewest extra unknowns.
- v = v₀ + aΔt
- Δx = v₀Δt + ½a(Δt)²
- v² = v₀² + 2aΔx
- Δx = ½(v₀ + v)Δt; all four assume constant a.
Avoid this: v_avg = (v₀ + v)/2 is valid for constant acceleration, not arbitrary motion.
Remember why: The equations summarize the same motion. Check that acceleration stays constant, choose a positive direction, then list knowns and the unknown before substituting.
Use one constant acceleration over the stated interval; recheck at a stop or model change.
Explain, don’t just substituteExplain why a correct equation can still give an inappropriate physical answer.
Refresh Kid · AP Physics 1 · Unit 1 · 1.3.A · 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 1.3. 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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