Imagine you're describing a car journey to a friend, but you can only use five pieces of information: where it started, how fast it started, how fast it ended up, how hard it accelerated, and how long that took. That's exactly what SUVAT gives you — five letters, five variables, and as long as acceleration stays constant, these five numbers completely describe the motion.

The five variables

Important: every one of those except t is a vector — meaning it can be positive or negative depending on direction. Get the signs wrong and your whole answer flips upside down (literally, sometimes).

How to pick the right equation, every time

This is the single most useful skill in this topic. Follow this 3-step routine and you'll never freeze on a SUVAT question again:

1. List every value the question gives you, using the correct symbol (s, u, v, a, t).
2. Identify which variable you need to find, and which one is not mentioned at all (not given, not asked for).
3. Choose the one equation out of the four that doesn't contain that missing variable.

Worked Example — Landing Plane

A pilot is landing a small aircraft. The airstrip is only 450 m long, and the aircraft decelerates from 40 m s⁻¹ at a constant rate of 2 m s⁻². Will the plane stop before it reaches the end of the runway?

Think of a motion graph as a "story" of a journey. A displacement-time graph tells you where something was at every moment; a velocity-time graph tells you how fast and in which direction; an acceleration-time graph tells you how quickly the speed itself was changing. The trick to reading any of them is remembering just two tools: the slope (gradient) and the area under the graph.

Displacement–Time Graphs

• Slope = velocity. Straight (diagonal) line → constant velocity. Curved line → acceleration.
• Positive slope = moving in the positive direction; negative slope = moving in the negative direction.
• Zero slope (horizontal line) = object is at rest.
• The area under a displacement-time graph means nothing physical — ignore it here.

Velocity–Time Graphs

• Slope = acceleration. Straight line = uniform acceleration; curved = non-uniform acceleration.
• Positive slope = speeding up in the positive direction; negative slope = speeding up in the negative direction (or slowing down, depending on context).
• Zero slope (horizontal) = constant velocity.
• Area under the line = displacement (or distance travelled). This is one of the most tested facts in this whole chapter.

Acceleration–Time Graphs

• Zero slope (horizontal line) = constant acceleration.
• Area under the line = change in velocity.
• The steepness of the slope itself is meaningless here (there's no "rate of change of acceleration" concept at this level).

Worked Example — Reading a Displacement-Time Graph

A runner sprints out to 50 m and back over 14 seconds, following a straight-line-up, straight-line-down displacement-time graph (peaking at 50 m around t = 7 s).

Non-Uniform Acceleration & Curved Lines

Sometimes acceleration itself changes — like a skydiver whose air resistance grows as their speed increases, eventually balancing their weight (terminal velocity). This produces a curved velocity-time graph. To find the instantaneous acceleration at one exact moment, you can't just use two random points — you need to:

1. Find the point on the x-axis matching the time you care about.
2. Draw a line straight up to meet the curve.
3. Draw a tangent to the curve at that exact point (a straight line that just "touches" the curve there).
4. Find the slope of that tangent as normal, using two points far apart on it.

Drawing Good Graphs

• Label both axes with quantity and unit (e.g. "distance / m").
• Space your scale evenly — use multiples of 2, 5, or 10. Never multiples of 3.
• Make your plotted data fill most of the graph paper — don't cram it into a corner.
• Plot points accurately to within half a small square, using a sharp pencil.
• Draw a line of best fit that balances points evenly above and below it, ignoring outliers (circle them instead).

Here's the one distinction that trips up more students than anything else in mechanics: the difference between distance and displacement, and between speed and velocity. Get this solid now and half of your future mechanics problems become instantly easier.

Scalars vs Vectors — the core idea

• Scalar = has magnitude only (a size, no direction). E.g. mass — "5 kg" doesn't point anywhere.
• Vector = has magnitude and direction. E.g. weight — it's a force, and it always points down.

The four key quantities

Other scalars & vectors you'll meet

Vector notation

Because it's easy to lose track of which quantities are vectors, physicists write them specially — either in bold italic (like F or s) in textbooks, or with a small arrow drawn over the top (like s with an arrow) when handwriting. That arrow does not point in the actual direction of the vector — it's just a label telling you "this quantity has a direction."

"Resolving" a vector just means breaking one diagonal vector into two perpendicular pieces — usually horizontal and vertical — that together have exactly the same effect as the original. This is the single biggest reason projectile motion becomes manageable: instead of dealing with one messy diagonal vector, you deal with two simple straight-line ones separately.

Two ways to resolve a vector

• By scale drawing: Draw the vector precisely to scale with a ruler and protractor, then measure the horizontal and vertical sides directly. Good for visualising, but slower and less accurate.
• By calculation: Sketch a rough (not-to-scale) triangle and use trigonometry — much quicker and more accurate once you're confident with it.

Resolving splits one vector into two. Adding vectors does the opposite — it combines two (or more) vectors into a single "resultant" vector that has the same overall effect. This resultant is sometimes called the "net" vector (e.g. net force).

Scale diagram methods

• Triangle method: Draw the vectors head-to-tail. The resultant runs from the tail of the first vector straight to the head of the last one.
• Parallelogram method: Draw the vectors tail-to-tail, complete the parallelogram shape, and the resultant is the diagonal running from the shared tail.

Calculation method

When two vectors are perpendicular (like a swimmer's velocity and a river's current), you don't need a scale drawing at all — just two clean steps:

Worked Example — Swimmer Crossing a River

A swimmer swims due north at 2 m s⁻¹. The current flows east at 5 m s⁻¹. Find the resultant velocity.

This is where everything from this chapter comes together. A projectile (like a thrown ball or a launched rocket) moves in two dimensions at once — but the golden rule that makes it manageable is: treat the horizontal and vertical motion completely separately. They only share one thing in common: time.

The two independent motions

The reason horizontal acceleration is zero is simple: once a projectile is released (ignoring air resistance), the only force acting on it is gravity — and gravity only pulls straight down, never sideways. So the horizontal velocity never changes throughout the whole flight.

Three scenarios you'll be tested on

• Vertical projection (free fall): something simply dropped — no horizontal motion at all.
• Horizontal projection: something launched sideways from a height (like a stunt bike leaving a ramp) — initial vertical velocity is zero.
• Projection at an angle: the full case — both horizontal and vertical initial velocities exist.

Worked Example — Horizontal Projection

A motorcycle stunt-rider moving horizontally takes off from 1.25 m above the ground, landing 10 m away. What was the take-off speed?

A free-body diagram strips away all the visual clutter of a real object and just shows the forces acting on it as clean, labelled arrows. This is one of those skills that feels simple but examiners specifically reward when done correctly — and penalise when arrows are missing, mislabelled, or the wrong length.

Three rules for drawing one

1. Rule 1: Draw a single point at the centre of mass of the object (you're modelling it as a particle).
2. Rule 2: Draw the object completely free from contact with anything else — no ground, no ramp, nothing touching it in the drawing.
3. Rule 3: Draw every force acting on it as a vector arrow — correct direction, and length roughly proportional to magnitude.

Objects in equilibrium — the closed triangle trick

If an object has exactly three forces acting on it and is in equilibrium (not accelerating), those three force vectors — when rearranged head-to-tail — will always form a closed triangle. This is a powerful way to solve problems: draw the forces on the free-body diagram, then redraw them as a triangle, and use trigonometry or Pythagoras on the triangle itself.

• I can list all five SUVAT variables and their units.
• I can identify the missing variable in a SUVAT problem and pick the correct equation.
• I know that u = 0 for "starts from rest" and v = 0 for "comes to rest" / top of a throw.
• I can read gradient and area correctly off displacement-time, velocity-time, and acceleration-time graphs.
• I can draw a tangent to a curved velocity-time graph to find instantaneous acceleration.
• I can explain the difference between distance/displacement and speed/velocity with an example.
• I can resolve a vector into horizontal and vertical components using sin and cos.
• I can add two vectors using both the triangle/parallelogram scale-drawing method and by calculation.
• I understand why horizontal velocity stays constant during projectile motion (ignoring air resistance).
• I can solve time of flight, maximum height, and range problems for a projectile launched at an angle.
• I can draw a correctly labelled free-body diagram for an object on a slope or hanging in equilibrium.
• I understand that three forces in equilibrium form a closed triangle.