Module 3: Kinematics · Method · Prerequisite: K1
Every physics book shows diagrams. Almost none teaches how to make one.
Mahavakya
A diagram is not a picture of the situation. It is a set of physical claims, and every arrow on it can be wrong.
You have seen hundreds of diagrams in solutions and been shown how to draw perhaps none of them. The skill is expected to arrive by osmosis, and for some students it does. For the rest, a diagram remains decoration — something drawn after the thinking, to accompany an answer, rather than the instrument that produced it.
Kinematics is where that stops working. A projectile problem is often solved the moment the axes are chosen. A constraint problem is solved once you have drawn which parts of the string change length. A relative-motion problem is a triangle. From here on, the drawing is the work.
On this page
1. A picture and a diagram are different things
A ball is thrown vertically upward at 20 m/s. Here are two drawings of that sentence.
The same sentence, drawn twice.
The left one is a picture. It shows what happens and supports nothing. There is no origin, so no position can be stated. There is no positive direction, so no sign can be assigned. There are no labelled vectors, so nothing can be substituted into an equation. It is a drawing of a memory.
The right one is a working diagram. It carries an origin, a positive direction, and every vector named with its value. From it you can write y = 20t − ½gt² without thinking again about which way anything points, because those decisions have already been made and recorded.
The test: can someone who has not read the problem write down the equations from your diagram alone? If not, it is a picture.
2. What a diagram commits you to
K1 said that motion is a relation between a body and a frame someone chose. A diagram is where that choice gets made, and it is made in three parts.
The origin. Mark it. Every position on the diagram is measured from it, and a problem where you forget which point is zero produces answers that are right by some other reckoning.
The axes and the positive direction. Draw the arrow and label it. Whether up is positive is your decision, and it does not matter which you choose — but it matters enormously that you fixed it before writing anything, and that you can see what you fixed.
The instant. A diagram shows one moment. The ball rising and the ball falling are different diagrams, and drawing them on top of each other is how students end up with a velocity and an acceleration that belong to different times.
Those three decisions cost about fifteen seconds and remove most of the sign errors students make in a whole paper. They are also the three things a picture leaves out.
3. An arrow is a claim
This is the heart of it.
When you draw an arrow labelled a pointing downward, you have asserted that the acceleration is downward. That is a physical statement, exactly as much as writing a = −9.8 m/s². If it is wrong, the diagram is wrong — and unlike a wrong formula, a wrong arrow is quiet. Nothing about it looks unusual, and the algebra that follows will be internally consistent and physically false.
Consider the top of a projectile’s flight.
Two claims about the same instant. One of them is false.
The left drawing is not careless. It is what a student produces after reasoning correctly that the ball stops rising — and then extending “stops” further than the physics allows. The vertical velocity is zero. The horizontal velocity is untouched, because nothing horizontal ever acted on it. And the acceleration was never zero for an instant: it is g downward from launch to landing.
Draw the left diagram and every subsequent line is doomed, however careful. Draw the right one and the answer is nearly written already.
An arrow you are unsure about is worth pausing on. If you cannot say why it points where it points, you have found the part of the problem you do not yet understand — before spending five minutes on algebra that was never going to work.
4. Drawing motion itself
So far the diagrams have shown one instant. There is a second kind that shows a whole motion, and it is the most useful drawing in kinematics that students are never taught.
Mark the position of the body at equal intervals of time — every second, say — and put a dot at each. Nothing else. No axes, no arrows, no numbers.
The spacing is the physics.
Equal spacing means equal distance in equal time, which is constant velocity. Spacing that grows means the body covers more ground each second, so it is speeding up. Spacing that shrinks means the reverse.
You have just read the velocity and the acceleration off a row of dots, without an equation, a graph or an axis. That is what a diagram can do that a formula cannot.
Look more closely at the middle row. The gaps go 1, 3, 5, 7 — the odd numbers. That is not decoration; it is what constant acceleration produces, and it was Galileo’s discovery. You will derive it in K8, and the dots are already showing it.
Use this drawing when a problem describes a motion in words and you cannot picture it. Six dots take ten seconds and turn a sentence into something you can look at. It is also the graph you will meet in K6, drawn before anyone has mentioned axes.
5. What to leave out
A good diagram is emptier than students expect. Its job is to carry the quantities the physics needs, and everything else competes for attention with the things that matter.
Leave out
- The shape of the object. A block is a dot or a small square. A car is a dot.
- The scenery — hills, trees, the person throwing the ball.
- Anything you have not been given and do not need.
Keep
- The origin and the positive direction.
- Every vector that acts, drawn from the body and labelled with its symbol.
- Every given distance and angle, marked where it applies.
- The unknown you are looking for, marked with a question mark.
That last item is worth adopting as a habit. A diagram with the unknown marked on it tells you what you are trying to find, and a surprising number of half-finished solutions are the result of forgetting.
6. Draw before you calculate
The order matters, and it is the reverse of what most students do.
Drawing after the algebra makes the diagram a summary — an illustration of a result you already have, which by then cannot help you. Drawing first makes it an instrument. The choices it forces you to make, about origin and direction and which instant you are describing, are precisely the choices that determine whether the algebra will work.
The sequence
- Read the problem once, without writing.
- Draw the situation as dots and arrows.
- Mark the origin and the positive direction.
- Label every known quantity where it belongs on the drawing.
- Mark the unknown.
- Now write the equations.
Steps two to five take under a minute. In an examination that minute buys back several, because the equations come out of the diagram rather than out of memory, and there is nothing left to second-guess about signs.
7. Reading your own diagram back
When an answer arrives, return to the drawing and ask whether the two agree.
You calculated a displacement of −12 m. Does the diagram show the body ending on the negative side? You found a velocity of 4 m/s upward. Does the arrow you drew point upward? You obtained a time of −3 s. Is there any moment on the diagram at which that could refer to something?
This check takes ten seconds and catches a category of error that no amount of re-checking the arithmetic will find — because the arithmetic is usually right. What went wrong was upstream, in a sign convention or a misread direction, and the diagram is the only place that decision is visible.
Contemplation
There is something odd about how much trust a diagram receives. A formula on the page invites scrutiny — students check it, hesitate over it, look it up. A drawing on the same page is believed immediately, and rarely questioned at all.
That is why a reversed arrow is more dangerous than a mistyped coefficient. The coefficient will be checked. The arrow will be absorbed, and it will quietly organise everything that follows — because a picture is understood before it is examined, and by then the mind has already accepted what it showed.
So a diagram deserves the same suspicion you would give a formula, and for the same reason: it is a claim about the world, made by someone who might be mistaken. Including when that someone is you.
Common misconceptions
1. The diagram is for the examiner
It is for you, and its main use is over before you write the first equation. Whether anyone else reads it is a separate matter.
2. A neater drawing is a better drawing
A rough sketch with the origin, the positive direction and every arrow labelled beats a careful drawing without them. Straight lines are not the point; stated commitments are.
3. Arrow length should be to scale
Only when you intend to read something off the drawing, as in a vector triangle. Otherwise the label carries the magnitude and the arrow carries the direction. Do not spend time making 20 m/s exactly twice as long as 10 m/s.
4. One diagram per problem
A diagram describes one instant. Motion in stages needs a diagram per stage, and problems involving two bodies often need one per body. Crowding several moments onto one drawing is how quantities from different times end up in the same equation.
5. Dots at equal spatial intervals show the motion
They show nothing. The whole content of a strobe diagram is that the intervals are equal in time. Equally spaced dots then mean constant velocity; equally spaced dots by construction mean only that you drew them that way.
6. If the answer is right, the diagram was right
Not necessarily. Two errors can cancel, and a diagram with a reversed arrow can still give a correct magnitude. The check is whether the diagram and the answer describe the same motion, not whether a number matched.
Reflection
- Take the last three problems you solved. Did you draw before or after the algebra? For which of the three would drawing first have changed anything?
- A friend hands you a diagram with no origin marked. What can you still work out from it, and what can you not?
- Why is a wrong arrow harder to notice than a wrong number?
Key takeaways
- A picture shows the situation. A diagram carries the origin, the positive direction and every labelled vector — enough to write the equations from.
- Every arrow is a physical claim, and a wrong one is quiet.
- A diagram describes one instant. Stages and second bodies get their own.
- Leave out shape and scenery. Keep origin, direction, given values, and the unknown.
- Draw before calculating, so the diagram is an instrument rather than a summary.
- Marking position at equal time intervals shows velocity and acceleration in the spacing, with no calculation at all.
- Read the answer back against the drawing. That catches errors the arithmetic cannot — and it catches your own errors, which simply looking at the diagram will not.
What comes next
From here the module is about motion itself, beginning with how fast something goes on average and how fast it is going right now — two questions that sound like one and are not. Every part that follows assumes you will draw first.
Prerequisite: K1 · Reading: 15 min · Practice: every problem from here on · Difficulty: Beginner