Module 3: Kinematics · Theory · Prerequisites: K4, Foundation 7
Everything K4 obtained by differentiating is visible on a graph without calculating anything.
Mahavakya
A graph is not a picture of the motion. It is the motion written in a second language, and some things are easier to say there.
On this page
Foundation 7 established that a slope is a rate and an area is an accumulation, and this part assumes it. What follows is that machinery applied to one particular graph, plus a question Foundation 7 never asked — which curves can represent a motion at all.
1. Height and slope are different things
On a position–time graph, two features carry two different quantities, and students conflate them constantly.
The height is the position. How far the curve sits above the axis is where the body is.
The slope is the velocity. How steeply the curve rises is how fast the body moves.
A body far from the origin may be perfectly still — high curve, flat slope. A body at the origin may be moving at speed — zero height, steep slope. The two features are independent, and asking about one tells you nothing about the other.
The height is the position. The slope is the velocity. They peak at different times.
The figure makes the point sharply. At the peak, the position is as large as it gets — and the tangent is horizontal, so the velocity is zero. The velocity is greatest much earlier, where the curve is steepest and the position is only middling.
“Maximum” applies to whichever quantity you named. A maximum of x is a place where the velocity vanishes. Students read the word “maximum” and assume the physical quantity in front of them is maximised — which is how a question about greatest height gets answered with a greatest speed.
2. Reading velocity off the curve
Four cases cover almost everything.
- Horizontal. Slope zero, so v = 0. The body is at rest — not at the origin, just not moving.
- Straight and rising. Constant positive slope, so constant positive velocity.
- Straight and falling. Constant negative velocity: the body moves steadily in the negative direction.
- Curved. The slope changes, so the velocity changes, so there is acceleration.
The third case is worth dwelling on. A falling line does not mean the body is slowing down; it means the body is moving backwards. Slowing down is a statement about the magnitude of the slope, and a line that is straight has a slope of constant magnitude however steeply it descends.
3. Curvature and acceleration
If the slope is the velocity, then the way the slope changes is the acceleration — and that is the curvature.
concave up → slope increasing → a > 0
concave down → slope decreasing → a < 0
Note that this says nothing about speeding up. A curve concave down with a negative slope is getting steeper downward — the velocity is becoming more negative, so the speed is increasing. That is K4’s va rule seen geometrically: same sign, speeding up, whatever the picture looks like at first glance.
And a straight line, of any steepness, has zero curvature and therefore zero acceleration. Two straight lines of very different slopes both describe unaccelerated motion.
4. Three independent quantities
Knowing where a body is tells you nothing about how fast it is going. Knowing how fast it is going tells you nothing about whether that is changing.
x = 10 m — the body is ten metres from the origin. It could be at rest, or moving either way at any speed.
v = +5 m/s — it is moving in the positive direction at five metres per second. It could be speeding up, slowing down, or neither.
a = −2 m/s² — the velocity is decreasing. Nothing here says where the body is or which way it is going.
Three separate pieces of information, and a single graph keeps them distinct: the height, the slope, and the curvature. That is most of why the graph is worth drawing.
5. Riding with the particle
Here is a way of reading a graph that works better than any list of rules.
Put yourself on the curve at t = 0 and travel along it, asking four questions continuously:
- Where am I? — the height.
- Which way am I moving? — the sign of the slope.
- How fast? — the steepness.
- Am I speeding up or slowing down? — whether the steepness is growing, and in which direction.
Run that along a curve and you have described the entire motion in words, without computing anything. It also tells you immediately when something interesting happens — the moment the slope changes sign is a reversal, and the moment the curvature changes sign is where the acceleration reverses.
Worked example
A particle’s position–time graph consists of three straight segments: from (0, 0) to (2, 6); then horizontal to (5, 6); then down to (8, −3). All distances in metres, times in seconds. Describe the motion.
0 to 2 s. Slope = 6/2 = +3 m/s. Moving forward at a steady 3 m/s, no acceleration.
2 to 5 s. Slope = 0. At rest, six metres from the origin. Note it is not at the origin.
5 to 8 s. Slope = (−3 − 6)/3 = −3 m/s. Moving backwards at 3 m/s, passing the origin on the way at t = 7 s.
Total distance 6 + 0 + 9 = 15 m; displacement −3 m. Every segment is straight, so the acceleration is zero throughout — except at the two corners, where the velocity changes abruptly. Real motion cannot do that, which is the subject of the rest of this part.
6. Which graphs are possible
Here is a question Foundation 7 did not ask. Given an arbitrary curve, could it represent the position of a body against time at all?
Four questions settle it, and they are worth asking in this order.
The last one looks the most exotic and is the only one that could describe a real motion.
1. Is it single-valued? A vertical line must meet the curve at most once. If it meets twice, the body is in two places at one instant, and the curve is not a position function at all.
2. Is it continuous? A jump in position means the body was in one place and then, with no time elapsing, somewhere else. That is teleportation.
3. Is the slope finite? A vertical tangent means infinite velocity — which would require infinite energy. Note this is a different failure from a vertical segment: a segment fails the first test, a tangent passes it and fails this one.
4. Is the slope smooth? A corner means the velocity changes instantaneously. Unlike the first three this is not impossible — it is an idealisation, and a useful one.
That last point deserves care, because it is often stated wrongly.
A corner on an x–t graph is what a textbook draws for a ball bouncing off a wall. The collision really takes a very short time, during which a very large acceleration acts — and the corner is the model that ignores that interval.
So a corner is not forbidden. It is an approximation, and it hides an impulse. The first three failures are impossible; the fourth is a modelling choice.
7. Unusual is not impossible
Look again at the four panels. The rejected curves are the ones that look ordinary — a smooth arc, a couple of horizontal lines, a rising curve. The accepted one is the wave, which looks the strangest of the four.
That is the point worth carrying away. Visual smoothness is not a test of physical validity. A closed loop can be drawn perfectly smoothly and still put a body in two places at once. A sine curve looks exotic and describes something entirely ordinary — a body oscillating back and forth, which is most of what mechanical systems do.
So do not judge a graph by whether it looks like the graphs you have seen. Ask the four questions.
A worked case: x² + t² = R²
Can a circle represent a motion? Solving gives x = ±√(R² − t²), and the ± is fatal: for every t between −R and R there are two positions. The full circle fails the first test.
But take only the upper branch, x = +√(R² − t²). Now it is single-valued and continuous. It still has vertical tangents at t = ±R, so it fails the third test at those two points — but everywhere strictly between, it is a perfectly good position function. The right conclusion is not “a semicircle is impossible” but that a relation must assign one position to each instant.
Contemplation
Notice what the four questions actually are. Single-valued, continuous, differentiable, smooth — these are conditions on a function. They come from mathematics, not from physics. Nobody performed an experiment to discover that a body cannot be in two places at once; it is built into what we mean by position.
And yet applying them rules out motions that are physically impossible, and permits ones that are physically ordinary. The mathematics turns out to have been shaped, long before anyone drew a position–time graph, by the same constraints the world obeys.
Foundation 1 said that mathematics does not govern motion but records the regularities we observe. This is a case where the recording is so faithful that the mathematical conditions and the physical ones become hard to tell apart — and it is worth remembering that they are not the same thing, even here.
Common misconceptions
1. A high point on the graph means a fast body
Height is position. Speed is slope. The highest point of a smooth curve is where the velocity is zero.
2. A falling line means slowing down
It means moving backwards. A straight falling line has constant speed; slowing down would show as the line flattening.
3. A curve going upward means acceleration
Rising is about the sign of the slope; acceleration is about whether the slope changes. A straight rising line has no acceleration at all.
4. A vertical segment just means very fast
It means two positions at one instant, which is a different and worse failure. A vertical tangent is the “very fast” case, and it is impossible for its own reason.
5. A corner is impossible
It is an idealisation. Textbooks draw corners for collisions, and what they hide is a brief interval of very large acceleration. Impossible and idealised are not the same objection.
Reflection
- Sketch an x–t graph for a ball dropped onto a hard floor and bouncing twice. Where are the corners, and what does each one hide?
- Can a position–time graph have a horizontal tangent without the body reversing? Draw one.
- The four tests are conditions on a function. Which of them would a photograph of a real motion ever violate?
Key takeaways
- Height is position, slope is velocity, curvature is acceleration — three independent readings from one curve.
- Where the position is greatest the velocity is zero, not greatest.
- A falling line means backward motion, not slowing down.
- Ride along the curve asking where, which way, how fast, and is that changing.
- Four tests decide whether a curve can be a motion: single-valued, continuous, finite slope, smooth slope.
- The first three failures are impossible. A corner is an idealisation hiding an impulse.
- Visual smoothness is not physical validity. Ask the questions.
What comes next
The position–time graph gives velocity by its slope and acceleration by its curvature — and curvature is hard to judge by eye. Plot the velocity directly and the acceleration becomes a slope again, which is far easier to read. K6 takes the velocity–time and acceleration–time graphs, where a second quantity also becomes available: the area.
Prerequisites: K4 and Foundation 7 · Reading: 15 min · Practice: 25 min · Difficulty: Intermediate