Five letters — s, u, v, a, t — run every constant-acceleration problem in physics: displacement, initial velocity, final velocity, acceleration, and time. Know any three and the other two are locked in by the SUVAT equations. This SUVAT calculator handles the two setups students and engineers meet most: starting velocity with acceleration over time, and starting and final velocities with acceleration. It returns the missing quantities with the exact equation used for each, so the working is visible, not hidden.
The Five SUVAT Equations
For constant acceleration, five equations connect the five quantities, each one omitting a different variable: v = u + at (no s), s = ut + ½at² (no v), s = vt − ½at² (no u), v² = u² + 2as (no t), and s = ½(u + v)t (no a). The exam-room skill is choosing the equation whose four letters are your three knowns plus your unknown. This calculator’s two modes cover the most common choices, and its result line names the equation it used so you can follow the pattern.
Worked Example 1: The Dropped Stone
A stone drops from rest (u = 0) under gravity (a = 9.8 m/s²) for t = 5 s. Final velocity: v = 0 + 9.8 × 5 = 49 m/s — about 176 km/h. Displacement: s = 0 + ½ × 9.8 × 25 = 122.5 m. The calculator’s average-velocity check agrees: the average of 0 and 49 is 24.5 m/s, times 5 seconds is the same 122.5 m. Two independent routes landing on one answer is how you know the setup was right.
Worked Example 2: The Braking Car
A car traveling at u = 30 m/s brakes at a = −6 m/s² to a stop (v = 0). Mode 2: time t = (0 − 30) ÷ (−6) = 5 s, and stopping distance s = (0 − 900) ÷ (2 × −6) = 75 m. Negative acceleration is not a special case — it is the same equations with a signed a, and the calculator handles the signs as long as you keep one direction positive throughout the problem.
The One Assumption: Constant Acceleration
SUVAT is exact only when a does not change. Free fall near Earth’s surface qualifies (ignoring air resistance). A car launching or braking at a steady rate qualifies. A car in real traffic, a rocket burning off fuel mass, a parachutist approaching terminal velocity — these have varying acceleration, and SUVAT returns only a rough average-based estimate for them. When a problem gives you three of the five letters and a story about steady speeding up or slowing down, it is telling you the assumption holds.
Sign Conventions: Where Most Errors Live
Pick a positive direction and never renegotiate mid-problem. Toss a ball upward with up positive: u is positive, a is −9.8 (gravity points down), and at the top v = 0 while s is the peak height. Drop the same ball with down positive and every sign flips — both approaches give identical magnitudes. Mixing conventions (positive u, positive a for gravity, upward displacement counted positive) is responsible for more lost marks than the algebra ever is. The calculator takes the numbers exactly as signed; your job is the consistency.
Common Mistakes
Reaching for SUVAT when acceleration varies is the conceptual error. The mechanical ones: using v² = u² + 2as with v and u swapped (the equation is not symmetric — final minus initial), squaring the sum instead of summing the squares, and forgetting that s in these equations is displacement from the start point, so a round trip home ends at s = 0 however far the odometer turned. Finally, unit discipline: SUVAT wants metres, seconds, and metres per second squared as a family — convert km/h to m/s (divide by 3.6) before you begin.
A Word on Graphs
Every SUVAT equation is a graph in disguise. On a velocity-time plot, acceleration is the slope and displacement is the area underneath — which is why s = ½(u + v)t works: that area is a trapezium. When an equation’s letters feel arbitrary, sketch the v-t graph and the formula usually names itself.
Related Tools
More mechanics tools are in the Physics Calculators hub, including the Projectile Motion Calculator and the Newton’s Second Law Calculator.
Frequently Asked Questions
What does SUVAT stand for?
The five quantities of constant-acceleration motion: s displacement, u initial velocity, v final velocity, a acceleration, t time. Know three, and the equations determine the other two.
Which SUVAT equation should I use?
List your three knowns and your unknown; choose the equation containing exactly those four letters. v = u + at when s is absent, v² = u² + 2as when t is absent, s = ½(u + v)t when a is absent.
How far does a dropped object fall in 5 seconds?
s = ½ × 9.8 × 5² = 122.5 m, reaching 49 m/s — this page’s first worked example, ignoring air resistance.
Can SUVAT handle deceleration?
Yes — enter acceleration as a negative number with a consistent sign convention. Braking from 30 m/s at −6 m/s² stops the car in 5 s over 75 m.
When is SUVAT not valid?
Whenever acceleration changes during the motion: real launches with throttle variation, rockets losing mass, objects near terminal velocity, or anything driven by a varying force. Then only calculus or numerical methods are exact.
Why does one equation have no t in it?
v² = u² + 2as links the velocities, acceleration, and displacement directly — it is the work-energy theorem in disguise, and it is the right tool when time is neither known nor asked for.
What units does the SUVAT calculator use?
Metres, metres per second, metres per second squared, and seconds. Convert kilometres per hour to metres per second first by dividing by 3.6.