The SUVAT equations are five equations of motion that describe an object moving with constant (uniform) acceleration, linking displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). Each equation leaves out one of these variables, so you choose the one that matches the quantities you know and the one you want.
Press the accelerator and your car doesn’t jump straight to 60 — it climbs there, second by second. Hit the brakes and it doesn’t stop dead; it slides to a halt over a distance you’d rather not underestimate.
Every one of those moments — speeding up, slowing down, a ball arcing through the air — obeys the same five compact formulas. Physicists call them the SUVAT equations, and once you know them, motion stops being a mystery and becomes something you can predict with a pencil.
What Are the SUVAT Equations?
SUVAT is a memory aid, not a Latin word. Each letter stands for one of the five quantities that describe an object moving in a straight line with steady acceleration:
- s — displacement: how far it moves, and in which direction (metres).
- u — initial velocity: the speed it starts with (metres per second).
- v — final velocity: the speed it ends with (metres per second).
- a — acceleration: how quickly the velocity changes (metres per second squared).
- t — time: how long the motion lasts (seconds).
Here is the precise version. The SUVAT equations are a set of five formulas that connect these quantities for any object whose acceleration stays constant throughout the motion. Hand the equations any three of the five values, and they give you the other two.
That last sentence hides the whole trick. A typical question quietly tells you three of the quantities and asks for a fourth — so the real skill is spotting which three you already have.
The five SUVAT quantities in one picture: an object starts at velocity u, accelerates at a, and reaches velocity v after travelling a displacement s in time t.
In practice, students lose more marks to muddled units than to muddled algebra. Convert everything to metres, seconds, and metres per second before you substitute, and most errors vanish.
The Five SUVAT Equations
There are five SUVAT equations, and each one leaves out a different variable. That is precisely why there are five: whichever quantity you don’t know, one equation doesn’t need it.
Linking final velocity, initial velocity, acceleration and time (no displacement):
Linking displacement to initial velocity, acceleration and time (no final velocity):
Linking displacement to both velocities and time (no acceleration):
Linking the two velocities to acceleration and displacement (no time):
And linking displacement to final velocity, acceleration and time (no initial velocity):
What each symbol means
- s = displacement — metres (m)
- u = initial velocity — metres per second (m/s)
- v = final velocity — metres per second (m/s)
- a = acceleration — metres per second squared (m/s²)
- t = time — seconds (s)
Which SUVAT equation should you use?
The fastest method: write down what you know, mark what you want, then pick the equation that contains those quantities — and skips the one you neither have nor need.
| Equation | Links these | Leaves out | Use it when you don’t know… |
|---|---|---|---|
| v = u + at | u, v, a, t | s | displacement |
| s = ut + ½at² | s, u, a, t | v | final velocity |
| s = ½(u + v)t | s, u, v, t | a | acceleration |
| v² = u² + 2as | s, u, v, a | t | time |
| s = vt − ½at² | s, v, a, t | u | initial velocity |
One row earns its keep more than the rest. When a problem gives you no time and asks for none — “how far to stop?”, “how fast at the bottom?” — reach for v² = u² + 2as.
How the SUVAT Equations Work
Where do five equations come from? Not from on high. From a single straight line on a velocity–time graph.
When acceleration is constant, a graph of velocity against time is a straight line. It begins at the initial velocity u and climbs (or falls) steadily to the final velocity v. Two features of that line give us everything.
For constant acceleration the line is straight. Its gradient is the acceleration a; the shaded area beneath it is the displacement s — a trapezium of area ½(u + v) × t, which is exactly s = ½(u + v)t.
First, the gradient. The steepness of the line is how fast velocity changes per second — that is the acceleration. Rearrange “gradient = (v − u) / t” and the first SUVAT equation, v = u + at, falls straight out.
Second, the area under the line. On any velocity–time graph, the area beneath the curve equals the displacement. Here that area is a trapezium with parallel sides u and v and width t.
A trapezium’s area is its average height times its width — ½(u + v) × t. That is the equation s = ½(u + v)t, read directly off the geometry.
The remaining three equations are algebra, not new physics. Substitute v = u + at into s = ½(u + v)t and you get s = ut + ½at². Eliminate the time between those two, and the time-free equation v² = u² + 2as appears. The full step-by-step working is laid out clearly in The Physics Hypertextbook.
One quiet assumption does all the work: that the average velocity sits exactly halfway between u and v. That is true only when acceleration is constant — a neat result first written down at Merton College, Oxford, back in 1335.
Change the starting velocity, the acceleration and the clock below, and watch all five quantities update together.
When Can You Use the SUVAT Equations?
The SUVAT equations are powerful, but they come with strict conditions. Break one and the answers quietly turn wrong.
Acceleration must be constant. This is the headline rule. If the acceleration changes during the motion, no single SUVAT equation describes the whole journey. The fix is simple: split the motion into stages where the acceleration is constant, and apply SUVAT to each stage in turn.
Motion must be in a straight line. SUVAT is one-dimensional. For two-dimensional motion such as projectile motion, you split the motion into horizontal and vertical directions and apply SUVAT to each axis separately.
Keep your units consistent. Use metres, seconds, metres per second and metres per second squared throughout. A speed given in km/h or a distance in centimetres must be converted first.
Mind the signs. Velocity and acceleration are vectors, so direction matters — which is part of the difference between velocity and speed. Choose one direction as positive, then anything pointing the other way (a deceleration, or gravity on a rising ball) takes a negative sign.
These conditions, with many fully worked cases, are covered in OpenStax College Physics. As a sanity check: a family car pulling away manages roughly 2–4 m/s²; if your answer says 50 m/s², something is off.
Real-World Examples of the SUVAT Equations
The equations aren’t just exam fodder. They quietly run through dozens of everyday situations.
A dropped object. Let go of your phone and it falls under gravity alone, so a = g ≈ 9.81 m/s² downward and the initial velocity u is zero. With those two values, s = ut + ½at² tells you how far it has fallen after any time t.
Braking a car. Here v = 0 when the car stops. Because v² = u² + 2as, the stopping distance grows with the square of the speed — double your speed and you need roughly four times the distance to stop. That single equation is the physics behind every speed-limit sign near a school.
An aircraft taking off. A jet must reach a minimum lift-off speed before the runway runs out. Knowing its acceleration and that target velocity, v² = u² + 2as gives the runway length required — which is why heavier loads need longer runways.
A ball thrown upward. At the very top of its flight the ball is momentarily still, so v = 0 — the trick that lets you find the maximum height. On the way down the same equations run in reverse, with gravity now speeding it up.
Common Misconceptions About the SUVAT Equations
A handful of misunderstandings trip up almost every learner. Clear these and SUVAT becomes far more reliable.
“SUVAT works for any motion.” It doesn’t. The equations assume constant acceleration. If the acceleration itself changes with time, you need calculus, not SUVAT — or you break the motion into constant-acceleration stages.
“Deceleration needs a different formula.” No. Slowing down is simply a negative acceleration in the very same five equations. There is no separate “deceleration equation” — just substitute a negative value of a once you’ve fixed a positive direction.
“g is always −9.81.” The sign of gravity depends on your choice of positive direction. If you call upward positive, then g = −9.81 m/s². If you call downward positive, g = +9.81 m/s². Both are correct; what matters is being consistent within the problem.
“Average velocity is always halfway between the two.” Only when acceleration is constant. The formula ½(u + v) for average velocity is a special case, not a universal truth — it’s exactly the assumption that makes SUVAT work.
How SUVAT Connects to Newton’s Laws, Energy and Projectiles
SUVAT doesn’t sit in isolation. It’s one node in a web of mechanics, and seeing the links makes each part stick.
It pairs with force. SUVAT tells you how a constantly accelerating object moves, but not why it accelerates. That answer comes from Newton’s second law, F = ma. A common workflow is to find the acceleration from the forces, then feed that a straight into a SUVAT equation. The broader picture of forces and motion lives in Newton’s laws of motion.
It echoes energy. Multiply v² = u² + 2as by ½m and rearrange, and you get ½mv² − ½mu² = (ma)s — the change in kinetic energy equals force times distance. The kinematic equation and the work–energy theorem are two views of the same motion.
It extends to two dimensions. Throw a ball at an angle and SUVAT still works — you simply apply it twice, once to the horizontal motion and once to the vertical. That is the whole basis of projectile analysis.
And when acceleration genuinely varies? Calculus takes over, with integration generalising the same ideas. SUVAT is the constant-acceleration corner of a much larger toolkit.
