The SUVAT Equations: Solve Any Constant-Acceleration Problem

Almost every motion problem in introductory physics is a SUVAT problem. If acceleration is constant, five variables — displacement s, initial velocity u, final velocity v, acceleration a and time t — are tied together by five equations. Know any three, and the other two are fixed. The only skill is picking the right equation, and there’s a simple trick for that.

The five equations

Here they are, with the variable each one doesn’t contain:

Equation	Missing variable
v = u + at	s
s = ½(u + v)t	a
v² = u² + 2as	t
s = ut + ½at²	v
s = vt − ½at²	u

That “missing variable” column is the whole trick.

How to pick the right equation

The reliable method:

1. Write down the three values you know and mark the one you want to find — that’s four of the five variables.
2. The fifth variable is the one you neither know nor need.
3. Choose the equation that leaves out that fifth variable.

Example. A car accelerates from u = 0 at a = 2 m/s² for t = 5 s. How far does it travel?

• Known: u, a, t. Want: s. Not involved: v.
• The equation missing v is s = ut + ½at².
• s = 0 × 5 + ½ × 2 × 5² = 25 m.

No guessing — the missing-variable rule points straight to the equation. The SUVAT calculator does this selection for you: enter any three values and it solves the rest.

The square-root one to watch

v² = u² + 2as is the equation with no time in it — perfect for “how fast after this distance” questions. But it hides a trap: solving for v (or u) needs a square root, and taking a root gives two answers, + and −. Physically that’s an object passing a point moving forwards or backwards. Pick the sign that matches your chosen positive direction. (This is also exactly the algebra that AI chatbots are documented to get wrong — a deterministic solver won’t.)

SUVAT and gravity

Vertical motion under gravity is a SUVAT problem because gravity gives a constant acceleration g ≈ 9.8 m/s². So:

• Free fall — drop something and use a = g. The free-fall calculator is SUVAT with u = 0.
• Projectile motion — the vertical component is SUVAT with a = −g; the horizontal component has a = 0 (constant velocity). The projectile calculator runs both and plots the path.

Just be consistent with signs: if you call up positive, then g is −9.8 and downward displacement is negative.

When SUVAT doesn’t work

The one hard rule: acceleration must be constant. If it changes — air resistance that grows with speed, a varying applied force, circular motion — SUVAT is invalid, and you need calculus (integrating the acceleration) instead. For the constant-acceleration world of most exam questions, though, the five equations cover everything.

Quick summary

The SUVAT equations describe constant-acceleration motion with five linked variables (s, u, v, a, t). Know any three and solve the other two by choosing the equation that omits the variable you neither know nor need. Watch the ± from the square root in v² = u² + 2as, use a = g for gravity, and remember it only holds when acceleration is constant. Solve one instantly with the SUVAT calculator.

Sources: standard kinematics (equations of motion for uniform acceleration) as taught in GCSE/A-Level/AP/IB physics. Educational information.