The 5 SUVAT equations, and exactly when to use each one
SUVAT equations connect displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t) for motion in a straight line with constant acceleration. This guide gives you all five equations, when to reach for each one, and worked examples so you stop guessing which formula fits.
What does SUVAT actually stand for?
SUVAT is an acronym for the five variables that describe straight-line motion under constant acceleration: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). Each of the five SUVAT equations links four of these variables, letting you solve for whichever one you don’t already know.
SUVAT problems only work when acceleration is constant — a car braking steadily, an object in free fall, a ball rolled up a slope. The moment acceleration changes mid-problem (like a rocket burning fuel), you need calculus instead.
These equations are a direct algebraic shortcut for calculus-based kinematics. If you already know calculus, you could derive every SUVAT equation yourself by integrating a = dv/dt — but SUVAT lets you skip straight to the answer.
The 5 SUVAT equations
Each one connects four of the five variables — pick the equation that already contains the variable you don’t know.
| Concept | Formula | What it means | Real-world example |
|---|---|---|---|
| Velocity-time | v = u + at | Final velocity from initial velocity, acceleration and time | How fast a dropped ball is falling after 2 seconds |
| Displacement (avg. velocity) | s = ½(u + v)t | Displacement using the average of initial and final velocity | Distance covered by a car accelerating steadily from a stop light |
| Displacement (initial velocity) | s = ut + ½at² | Displacement without needing the final velocity | How far a ball travels in the first 3 seconds of free fall |
| Velocity-squared | v² = u² + 2as | Final velocity when time isn’t known or needed | Impact speed of an object dropped from a known height |
| Displacement (final velocity) | s = vt − ½at² | Displacement using final velocity and time instead of initial | Working backward from a known landing speed |
How to pick the right equation
Look at which variable is missing from the question — that tells you exactly which SUVAT equation to use.
01 Missing s (displacement)?
Use v = u + at. This is the one to reach for when the question only asks about speed and doesn’t care about distance.
02 Missing v (final velocity)?
Use s = ut + ½at². Perfect for free-fall and projectile problems where you know the start speed but not the end speed.
03 Missing t (time)?
Use v² = u² + 2as. The go-to equation whenever a problem gives you a distance and asks for a speed, with no mention of time.
Where SUVAT equations show up in real life
- Car braking distances: Highway codes calculate stopping distances using v² = u² + 2as, working backward from a safe final velocity of zero.
- Free-fall physics: Anything dropped near Earth’s surface accelerates at 9.81 m/s², making it a textbook SUVAT scenario.
- Sports analytics: Sprinters’ acceleration off the blocks is modelled with SUVAT to estimate 100m split times.
- Elevator and escalator design: Engineers use s = ut + ½at² to keep acceleration comfortable for passengers over a fixed distance.
- Roller coasters: Designers calculate launch speed and braking zones using the same five equations.
- Rocket staging (early phase): Before fuel burn significantly changes mass, early rocket ascent can be approximated with constant-acceleration SUVAT.
Common SUVAT mistakes to avoid
- Mixing up sign conventions: Pick ‘up’ or ‘forward’ as positive at the start of a problem and stick with it — deceleration is just a negative acceleration, not a different formula.
- Forgetting acceleration must be constant: SUVAT breaks down the moment acceleration changes mid-motion, such as a car that brakes then accelerates again.
- Using the wrong equation when one variable is genuinely unknown: Don’t guess — identify exactly which of the 5 variables is missing, then match it to the equation without that variable.
- Ignoring units: Mixing km/h with m/s² is one of the most common sources of wrong answers — convert everything to SI units first.
- Assuming initial velocity is always zero: Only true for ‘starts from rest’ problems — always re-check the question for a non-zero starting speed.
Key takeaways
- SUVAT connects five variables — s, u, v, a, t — describing motion with constant acceleration.
- There are 5 equations; each one omits exactly one variable, so pick the equation missing whatever you don’t know.
- SUVAT only applies when acceleration is constant — variable acceleration needs calculus instead.
- v = u + at is the most-used equation when displacement isn’t involved.
- Sign convention matters: define a positive direction before you start substituting numbers.
- Real-world uses range from braking distances to free-fall and rocket ascent physics.
Frequently asked questions about SUVAT equations
What does SUVAT stand for?
SUVAT stands for the five variables in constant-acceleration motion: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time).
How many SUVAT equations are there?
There are 5 standard SUVAT equations, and each one relates four of the five variables, deliberately omitting one so you can solve for it.
When can’t you use SUVAT equations?
SUVAT equations only work when acceleration is constant. If acceleration changes over time — like a rocket burning fuel or a car with varying braking force — you need calculus-based kinematics instead.
What’s the difference between s = ut + ½at² and s = vt − ½at²?
Both find displacement, but the first uses initial velocity (u) and the second uses final velocity (v) — pick whichever variable the question actually gives you.
Do you need calculus to use SUVAT equations?
No. SUVAT equations are algebraic shortcuts already derived from calculus, so you can solve constant-acceleration problems with just algebra and a scientific calculator.
Ready to put SUVAT into practice?
Reading the equations is one thing — solving with them is another. Try the interactive SUVAT calculator to check your work step by step, or move on to projectile motion to see SUVAT applied in two dimensions.