Position, velocity, and acceleration
Relate an object’s position to instantaneous velocity and acceleration.
Position, velocity, and acceleration is one of 7 applications formulas in the calculus section of this library, and it is used at high school · ap level.
Why position, velocity, and acceleration works
Velocity is defined as the rate at which position changes and acceleration as the rate at which velocity changes, so each one is just the derivative of the line above it. Differentiating twice takes you from where an object is to how quickly its motion is changing.
What each symbol means
$s$ is position, $v$ velocity, $a$ acceleration, and $t$ time.
Position, velocity, and acceleration: when it holds
Keep units consistent; speed is $|v|$ and total distance may require splitting where $v$ changes sign.
When it stops applying
The chain breaks wherever the motion is not smooth. A bounce or a collision leaves the velocity with a sudden jump, so acceleration has no value at that instant and each stretch of the motion must be handled on its own.
Position, velocity, and acceleration: a worked example
If $s=t^3$, then $v=3t^2$ and $a=6t$.
The mistake to avoid
What people do: Reporting total distance travelled as the difference between the ending and starting positions.
Why it goes wrong: That is displacement. If the object turns around partway, the return trip cancels part of the outward trip and the total is understated.
Do this instead: Find every time the velocity changes sign, then add the sizes of the movements on each stretch separately.
Position, velocity, and acceleration: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $s$ is position, $v$ velocity, $a$ acceleration, and $t$ time.
- Check the conditions before substituting. Keep units consistent; speed is $|v|$ and total distance may require splitting where $v$ changes sign.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where position, velocity, and acceleration comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about position, velocity, and acceleration
Is speed the same thing as velocity?
No. Velocity carries a sign that records the direction of travel, while speed is its size and is never negative.
What does negative acceleration mean?
It means velocity is decreasing. If the object is already travelling in the negative direction, that actually makes it move faster.
How do I find when the object is momentarily at rest?
Solve for the times when velocity equals zero. Those instants are also the only places where the direction of travel can switch.
How do I recover position from acceleration?
Integrate twice. The initial velocity fixes the first constant and the initial position fixes the second.