Equations of Motion — SUVAT in One Dimension

Every straight-line problem with steady acceleration is described by just five quantities. Once you have any three of them, the other two are fixed — there is no freedom left. That is what makes these problems solvable: physics gives you a small set of equations that lock the five quantities together.

The five variables are remembered by the word SUVAT:

• $s$ — displacement (how far, and which way, in metres)
• $u$ — initial velocity (speed and direction at the start, in m/s)
• $v$ — final velocity (speed and direction at the end, in m/s)
• $a$ — acceleration (rate of change of velocity, in m/s²)
• $t$ — time (the duration of the interval, in seconds)

Notice that $s$, $u$, $v$ and $a$ are all vectors — in one dimension they each carry a + or − sign from your chosen convention. Only $t$ is a scalar. This is exactly why the sign convention from L01 matters here: get a sign wrong and the whole answer flips.

The five SUVAT variables are $s$ (displacement), $u$ (initial velocity), $v$ (final velocity), $a$ (acceleration) and $t$ (time). All except $t$ are vectors carrying a sign. Knowing any three lets you find the other two — but only when acceleration is constant.

Pause — copy the five SUVAT variables, their units, and the "list-knowns-first" habit into your book before moving on.

We just saw the five SUVAT variables and that any three fix the rest. That raises a question: where do the equations that link them actually come from — are they just memorised, or can we build them? This card answers it → we derive the first two straight from the definition of acceleration and the area under a v–t graph.

You do not have to take these equations on faith. The first one is just the definition of acceleration rearranged, and the second is the area under a velocity–time graph — the same area you met in Lesson 5. Building them this way makes them far easier to remember and trust.

Equation 1: $v = u + at$

Acceleration is the change in velocity per unit time: $a = \dfrac{v - u}{t}$. Multiply both sides by $t$ and rearrange:

In words: the final velocity is the starting velocity plus however much the acceleration has added (or removed) over the time. No displacement appears — this equation is the one to use when you do not know and do not need $s$.

Equation 2: $s = ut + \tfrac{1}{2}at^2$

On a velocity–time graph for constant acceleration, the line is straight, sloping from $u$ up to $v$. The displacement is the area under that line. Split the area into a rectangle (height $u$, width $t$) plus a triangle (base $t$, height $v-u = at$):

$v = u + at$ comes straight from $a = (v-u)/t$. $s = ut + \tfrac{1}{2}at^2$ is the area under the v–t graph (rectangle $ut$ + triangle $\tfrac{1}{2}at^2$). Both assume constant acceleration.

Pause — copy the two derivations (definition of $a$ → equation 1; v–t graph area → equation 2) into your book before moving on.

We just saw the first two equations built from a definition and a graph. That raises a question: what about problems that don't give you time, or don't give you the final velocity — is there an equation for those too? This card answers it → yes; there are five equations in total, and each one is the one that omits a different variable.

There are exactly five equations of uniformly accelerated motion. The clever part is that each equation is built to leave out one of the five variables. So the variable a problem does not mention is your signpost to the right equation.

Which variable does each leave out?

The five equations: $v=u+at$ (no $s$); $s=ut+\tfrac{1}{2}at^2$ (no $v$); $v^2=u^2+2as$ (no $t$); $s=\tfrac{1}{2}(u+v)t$ (no $a$); $s=vt-\tfrac{1}{2}at^2$ (no $u$). The variable a problem doesn't mention points you to the right equation.

Pause — copy all five equations and the "missing variable" label beside each one into your book before moving on.

We just saw that each equation omits one variable. That raises a question: in a real problem, how do you turn a wall of words into the right equation — and what happens when something slows down or falls? This card answers it → with a fixed routine for selecting the equation, plus the sign rules that keep braking and free-fall correct.

Solving a SUVAT problem is a routine, not a guess. List what you know, mark what you want, and the missing variable picks the equation. The one thing that trips students up is signs — so decide your positive direction first, and never change it mid-problem.

The four-step routine

1. State a sign convention. Choose one direction as positive (e.g. "forward = +", or "down = +" for a falling object).
2. List SUVAT. Write the five letters and fill in the three you know, each with its sign. Mark the unknown you want.
3. Pick the equation that contains your three knowns and your wanted unknown — i.e. the one missing the variable you don't have.
4. Substitute and solve, then check the sign and size of the answer make physical sense.

Routine: (1) state a sign convention, (2) list SUVAT with signs, (3) pick the equation missing the variable you don't have, (4) substitute, solve, sanity-check. For braking, $a$ is negative (opposes motion); for free fall, $|a| = g = 9.8$ m/s².

Pause — copy the four-step routine and the sign rules for braking and free fall into your book before moving on.

We just saw the routine for choosing an equation and setting signs. That raises a question: how does this play out on real exam-style problems — a braking car and a falling object? This card answers it → with two fully worked solutions you can copy as templates.

Time to put the routine to work. Watch how listing SUVAT first makes the equation choice almost automatic, and how the sign convention is stated before any number is substituted.

Template: list SUVAT, spot the missing variable, choose the equation. Braking (no $t$): $v^2=u^2+2as$ gives a negative $a$. Free fall: set $a=g=9.8$ m/s² in your sign convention, then $s=ut+\tfrac{1}{2}at^2$ and $v=u+at$.

Pause — copy both worked examples (the braking car and the free-fall ball) into your book as solution templates before moving on.