SUVAT Kinematics Calculator
Constant-acceleration kinematics (SUVAT)
Many 1D motion problems can be modeled as straight-line motion with constant acceleration. In that case, the motion is described by five variables: displacement s, initial velocity u, final velocity v, acceleration a, and time t. “SUVAT” is simply a mnemonic for these symbols. If you know any three of the five (in a physically consistent way), you can usually solve for the remaining two.
How to use this calculator
- Enter any three values (leave the other fields blank).
- Use a consistent sign convention. For example, choose “forward/up” as positive. Then a braking acceleration is negative, and downward gravity could be negative if “up” is positive.
- Keep units consistent (common choice: m, s, m/s, m/s²). If you use km/h or ft/s, convert everything to a consistent system first.
Definitions
s = displacement (m) — change in position along the line (can be negative)
u = initial velocity (m/s)
v = final velocity (m/s)
a = constant acceleration (m/s²)
t = elapsed time (s), typically reported as a non‑negative value
The SUVAT equations
Under constant acceleration, the following relationships hold:
These formulas are not independent; you can derive one from another. Practically, they provide multiple ways to solve for unknowns depending on which three quantities you know.
Interpreting the results
- Sign matters: A negative value of s, u, or v simply means the direction is opposite to your chosen positive direction.
- Braking vs accelerating: If v is smaller than u (in the same direction), then a should be negative.
- Multiple solutions: Some input combinations (notably when solving for time from s = ut + ½at²) can yield two mathematical time roots. Usually the physically meaningful one is the non‑negative time consistent with the scenario.
- No real solution: If the algebra implies a negative discriminant (when a quadratic is involved), the inputs are mutually inconsistent for constant-acceleration 1D motion.
Worked example
Example: A car starts at u = 0 m/s and accelerates at a = 2.5 m/s² for t = 8 s. Find v and s.
- Final velocity: v = u + at = 0 + (2.5)(8) = 20 m/s.
- Displacement: s = ut + ½at² = 0·8 + ½·2.5·8² = 1.25·64 = 80 m.
So the car reaches 20 m/s after 8 s and travels 80 m in that time.
Which equation should you use?
| Known | Often solve for | Convenient equation(s) |
|---|---|---|
| u, a, t | v, s | v = u + at; s = ut + ½at² |
| u, v, t | a, s | a = (v − u)/t; s = (u + v)t/2 |
| u, v, a | t, s | t = (v − u)/a; v² = u² + 2as |
| u, a, s | t, v | s = ut + ½at² (quadratic in t); then v = u + at |
| v, a, s | t, u | v² = u² + 2as (solve for u); then t = (v − u)/a |
Assumptions and limitations
- 1D motion only: Motion is along a straight line. For 2D/3D projectile motion you must treat axes separately (e.g., x and y components).
- Constant acceleration: If acceleration varies with time (engine power curves, drag forces, thrust profiles), SUVAT may not apply without approximation.
- Neglecting resistive forces (unless they’re baked into a constant a): Air resistance typically makes acceleration non-constant.
- Sign convention is your responsibility: The equations work with any consistent convention, but mixing directions (e.g., using positive s with negative u unintentionally) can produce confusing results.
- Quadratic time ambiguity: When solving s = ut + ½at² for t, there may be two roots. The physically meaningful root is usually t ≥ 0 and consistent with the rest of the inputs.
- Input consistency: Some combinations are over‑constrained (entering 4–5 values). If those values conflict, a solver may be unable to satisfy all equations simultaneously.