Derive the constant-acceleration equations
Start from the definitions of acceleration and average velocity to derive the equations for motion in a straight line with uniform acceleration.
AS Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 2. The falling-object activity is an online description and calculation practice; it does not provide supervised laboratory competence. Full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use SI units and signed values, and recall that velocity includes direction.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which definition begins the derivation of v = u + at?
What you will learn
- Derive v = u + at from the acceleration definition
- Derive displacement equations using average velocity and graph area
- State the constant-acceleration assumptions
Acceleration gives the first equation
For uniform acceleration, a = (v − u)/t, where u is initial velocity, v is final velocity and t is elapsed time. Rearranging gives v = u + at. The equation follows directly from the definition; it is not a separate rule to memorise without conditions.
Uniform acceleration means the velocity–time graph is a straight line. If acceleration changes, this constant-acceleration equation cannot generally be applied over the full interval using one value of a.
Graph area and average velocity give displacement
For constant acceleration, the velocity–time graph is a straight line, so its area is a trapezium. Thus s = (u + v)t/2. Substituting v = u + at gives s = ut + 1/2at²; substituting u = v − at gives s = vt − 1/2at². Eliminating t with t = (v − u)/a gives v² = u² + 2as.
Here s is displacement, not necessarily distance. Choose a positive direction and keep u, v, a and s signed consistently. The equations assume straight-line motion and constant acceleration for the interval.
Pause and explain
Which condition is required for the standard constant-acceleration equations?
Worked example
Derive v² = u² + 2as from v = u + at and s = (u + v)t/2.
Show the worked solution
- Rearrange the first equation to t = (v − u)/a.
- Substitute into s = (u + v)t/2: 2as = (u + v)(v − u).
- Use the difference of squares: (u + v)(v − u) = v² − u².
Answer v² = u² + 2as.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Starting from a = (v − u)/t, derive v = u + at and state the condition.
Give me a hint
Rearrange algebraically and name the motion assumption.
Compare my reasoning
- Multiply both sides by t: at = v − u.
- Add u to both sides.
- State that a is constant for the interval.
v = u + at, for constant acceleration.
Look for these in your work
- Rearrangement correct
- Uniform-acceleration condition stated
A body starts with velocity u and accelerates uniformly for time t. Use the average velocity to derive s = ut + ½at².
Give me a hint
For a straight-line velocity–time graph, average velocity is (u+v)/2.
Compare my reasoning
- Write s = (u + v)t/2.
- Substitute v = u + at.
- Simplify: s = [u + (u + at)]t/2.
s = ut + ½at².
Look for these in your work
- Average-velocity relation used
- Substitution and simplification correct
Explain why s = ut + ½at² is not guaranteed to hold if acceleration varies during the motion.
Give me a hint
Identify which step in the derivation relies on a straight velocity–time graph.
Compare my reasoning
- The derivation uses v = u + at with one constant a.
- It also uses the trapezium area/average-velocity relation for a straight-line v–t graph.
- Variable acceleration can make the velocity–time graph curved.
The derivation assumes constant acceleration, giving a straight velocity–time graph and v = u + at. If acceleration varies, one constant a does not describe the interval, so this equation is not generally valid.
Look for these in your work
- Relevant constant-a assumption identified
- Graph/derivation limitation explained
Common mistakes
- Using a constant-acceleration equation when acceleration changes
- Mixing distance with signed displacement
- Forgetting that the equations are one-dimensional
What can you explain now?
Which definition begins the derivation of v = u + at?
Compare with the explanation
a = (v − u)/t
Rearranging the definition of uniform acceleration gives the equation.
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Make it stick
Which definitions and graph relationships lead to the constant-acceleration equations, and what assumption limits them?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.