Intuition The one core ideaAll of SUVAT is a single sentence: if you know how fast something's speed is changing, you can add up the tiny changes to rebuild its whole journey. Everything below — position, velocity, acceleration, slope, area — is just vocabulary for that one act of adding-up.
This page assumes you have seen nothing . Before you touch a single SUVAT formula on the parent note , you must own the words and pictures below. We build them in an order where each one leans only on the ones before it.
Position is a number that tells you where an object is along a chosen line, measured from a chosen starting point (call it the origin, at 0
Picture a toy car on a long ruler laid on the floor. Pick a spot and call it 0 3 m
Intuition Why we need a direction, not just a distanceLook at the figure: marks to the right of 0 left are negative. This sign is the whole reason the topic uses the special word displacement instead of "distance". Distance can only grow; position (and its change) can go negative when you move left.
Displacement s change in position, so it carries a sign : positive if you ended up on the positive side, negative if the negative side.
If the car starts at position 1 m 4 m s = 4 − 1 = 3 m 4 1 s = − 3 m
Common mistake Displacement is not distanceWhy it feels the same: for a one-way trip they are equal.
Fix: go right 5 m 5 m distance travelled = 10 m displacement s = 0 s Vectors .
==t since we started the clock . We always start our clock at t = 0
The picture: a stopwatch. At the instant the car is at its starting mark, the watch reads 0 t = 0 t
A graph draws one quantity against another: for every value on the horizontal axis, a dot at the matching value on the vertical axis. Trace the dots and you get a curve — the story of how the two quantities relate.
The two graphs the topic lives on:
position–time (s t velocity–time (v t Velocity-time graphs .
We need graphs because the two magic words coming next — slope and area — are pictures on a graph , not abstract ideas.
The slope at a point on a curve is how much the vertical quantity changes for a tiny step sideways : rise divided by run. Steep uphill = big positive slope; downhill = negative slope; flat = zero slope.
Intuition Why slope is the heart of everythingOn a position–time graph, "how fast is position changing?" is exactly the steepness of the curve. That steepness has a name: velocity . So velocity is nothing more than the slope of the position graph. Look at the red tangent line in the figure — its steepness IS the speed at that instant.
Velocity = the slope of the position–time graph = how fast position changes, with a sign for direction. Moving right (positive direction) → positive velocity; moving left → negative velocity.
The topic uses two names for the same thing at two moments:
==u t = 0 initial velocity.
==v t final (current) velocity.
Intuition Why two letters, not oneA journey has a before and a now . Every SUVAT formula relates the two. Keeping u v u v
Acceleration a velocity –time graph = how fast velocity is changing each second. Units: metres per second, per second (m/s 2
Picture the velocity–time graph. If it climbs steadily, its slope is a fixed number — that fixed number is a a > 0 a < 0
a straight slanted line , its slope is the same everywhere — one single number a a not constant (like in Simple Harmonic Motion ), the line bends and SUVAT breaks.
Definition The derivatived t d s d t d s s t slope of the s t , written as a symbol so we can calculate with it. The little d
So the parent's two starting equations translate straight from the pictures we just built:
Why this tool and not plain arithmetic? Because slope is about an instant — an infinitely thin moment — and only the derivative notation captures "the steepness right here , not averaged over a fat chunk of time." This is the subject of Differentiation and Integration .
∫ add up infinitely many tiny slices . On a graph, the sum of all the thin slices under a curve is the area between the curve and the horizontal axis.
Intuition Why integration is "slope run backwards"Slope asks: given the journey, how fast? Integration asks the reverse: given how fast, what's the journey? If velocity is the slope of position, then position is the area under the velocity curve. That's why every SUVAT derivation ends with an integral — we know a v s s
Intuition Why the trapezium?Because a straight , so the region under it is a trapezium (a rectangle plus a triangle). Its area — base t u v 2 1 ( u + v ) t is the equation s = 2 1 ( u + v ) t
Definition The sign conventionBefore any numbers, choose which direction is positive . Everything pointing that way is + − s u v a
Common mistake Flipping gravity's sign mid-problemWhy it feels safe: "gravity is 9.8 Fix: if you chose up = positive , then downward gravity is a = − 9.8 m/s 2 + 9.8 Free fall and g and every axis of Projectile motion .
Derivative ds/dt notation
Integral = area under curve
Positive direction choice
Test yourself — you're ready for the parent note only if each reveal matches your own answer.
What does displacement s A sign (direction) — it can be negative or zero.
In one phrase, what is velocity on a position–time graph? The slope (steepness) of the curve.
In one phrase, what is acceleration on a velocity–time graph? The slope of the velocity curve.
What does the symbol d t d s The rate of change of s s t
What does the symbol ∫ Adds up tiny slices = the area under the curve.
If velocity is the slope of position, then position is the ___ of velocity. Area under the velocity–time graph.
Why must the velocity–time line be straight for SUVAT to work? Because a a
What shape is the area under a constant-acceleration v t A trapezium, area 2 1 ( u + v ) t
If you choose "up = positive", what sign does gravity get? Negative: a = − 9.8 m/s 2
Which two symbols name velocity at the start and at time t u v
Differentiation and Integration — the derivative (slope) and integral (area) machinery this page defines.Velocity-time graphs — where slope gives a s Vectors — why s , u , v , a Projectile motion — needs the sign convention per axis.Simple Harmonic Motion — the case where a Free fall and g — the sign-of-gravity worked example. 
