Five quantities describe motion under constant acceleration, and three equations connect them. Give any three of the five and the other two follow — which is the whole method, and why these are worth knowing cold. They are often called the SUVAT equations after the letters involved.
The five quantities and the three equations
| Symbol | Meaning | Unit |
|---|---|---|
| u | Initial velocity, at the moment timing starts | m/s |
| v | Final velocity, at the end of the interval | m/s |
| a | Acceleration, which must stay constant | m/s² |
| s | Displacement — how far it moved, and in which direction | m |
| t | Time taken | s |
The three equations, and the one quantity each leaves out:
| Equation | Missing | Use it when |
|---|---|---|
| v = u + at | s | You do not know or care about the distance |
| s = ut + ½at² | v | You do not know the final velocity |
| v² = u² + 2as | t | You do not know the time — often the quickest route |
The method is mechanical once you see it: write down the three you know and the one you want, then pick the equation that contains those four and not the fifth. There is nothing to be clever about.
A fourth relation, s = ((u + v)⁄2) t, is sometimes listed as well. It says displacement equals average velocity times time, which is only true because the acceleration is constant.
The condition everything depends on
The acceleration must be constant. Not “roughly constant” — constant. If it changes during the interval, none of these equations applies and you need calculus instead.
In practice that condition is met more often than you would expect, because objects falling freely near the Earth’s surface accelerate at a steady 9.81 m/s² as long as air resistance can be ignored. That is why so many questions are about dropped or thrown objects.
It is not met by a car in traffic, a parachutist after the canopy opens, or a mass on a spring. Questions that describe any of those are testing whether you noticed.
A worked example
A stone is dropped from a cliff and hits the water 3.0 s later. How high is the cliff, and how fast is the stone travelling on impact?
Take downwards as positive. Dropped means it starts at rest, so u = 0. Gravity gives a = 9.81 m/s², and t = 3.0 s. We know three, so we can find the other two.
Height. We want s and do not know v, so use the equation without v:
s = ut + ½at² = 0 × 3.0 + ½ × 9.81 × 3.0² = 0 + ½ × 9.81 × 9 = 44.1 m
Impact speed. We want v and know u, a and t, so:
v = u + at = 0 + 9.81 × 3.0 = 29.4 m/s
Worth checking against the third equation, which should agree: v² = u² + 2as = 0 + 2 × 9.81 × 44.1 = 865.2, and √865.2 = 29.4 m/s ✓
Note that only t is squared in the second equation. Squaring the whole of ½at, or squaring at and then halving, are both common and both wrong.
Where marks get lost
Not fixing a positive direction first. Write down which way is positive before you write anything else. If down is positive, then g is +9.81 and an upward throw has a negative u. Mixing conventions halfway through a question is fatal and hard to spot afterwards.
Confusing displacement with distance. Throw a ball straight up and catch it: it has travelled some distance, but its displacement is zero. These equations use displacement.
Assuming “dropped” and “thrown down” are the same. Dropped means u = 0. Thrown downwards means u is whatever you were told, and the answer will differ.
Forgetting that v = 0 at the top of a rise. An object thrown upwards has zero velocity at its highest point, not zero acceleration — gravity is still pulling on it the whole time. That instantaneous v = 0 is usually the extra piece of information a question is relying on you to supply.
Squaring only part of a term. In ½at², the t is squared and nothing else is.
Questions students actually ask
What does SUVAT stand for?
The five symbols: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. It is a memory aid rather than a formal name.
How do I choose which equation to use?
List the three quantities you know and the one you want. That is four of the five. Use the equation that leaves out the fifth — each equation is missing exactly one, which is what makes the choice unambiguous.
Can I use these equations for something thrown upwards?
Yes, as long as you keep one sign convention throughout. If up is positive then a = −9.81 m/s², and the object’s displacement is negative once it falls below its starting point.
What is the difference between speed and velocity here?
Velocity has a direction and can be negative; speed is just its size. These equations are about velocity, which is why a negative answer is meaningful rather than an error.
Do these work if the acceleration changes?
No. Constant acceleration is the condition the whole set is derived under. For changing acceleration you need calculus — differentiating and integrating between the quantities instead.
Related formulas
Newton’s second law · Projectile motion · The quadratic formula
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