Not always. In a directly proportional relationship, twice as big gives twice as big, and the graph is a straight line through the origin. A taxi with a four-euro starting fare fails: its line is straight but starts at four euros. A spring passes: F = C · u, with the spring constant C as the constant of proportionality.
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Twice as far by taxi is twice as expensive too, right? It feels logical, but it is not true.
With a directly proportional relationship between two quantities, two things always hold. If one becomes twice as big, the other also becomes twice as big. And the graph is a straight line through the origin. A taxi with a four-euro starting fare and two euros per kilometre does not meet that: five kilometres costs fourteen euros, ten kilometres twenty-four. The line is straight, but it starts at four euros. Then the constant of proportionality: the formula has the form y = a · x. For a spring, that is the elastic force, F = C · u, and you get the spring constant C from the graph. Pick a point on the line that lies exactly on a grid line, as far from the origin as possible. At 8.0 N and 0.20 m, C = 8.0 / 0.20 = 40 N/m.
For high school physics, in the chapter on forces. Halfway through there is a question and a pause: stop the video there and work it out yourself first.
In this video
The full explanation, in writing
The taxi
0:00Twice as far by taxi is twice as expensive too, right? With a directly proportional relationship between two quantities, two things always hold. If one quantity becomes twice as big, the other also becomes twice as big. And the graph in which you plot them against each other is a straight line through the origin, the point where both are zero.
0:26The taxi doesn't meet that. Say: four euros to get in and two euros per kilometre. Five kilometres then costs fourteen euros, ten kilometres twenty-four. That is more, but not twice as much. The line is straight, but it starts at four euros and not at zero.
0:47So the price is not directly proportional to the distance.
The constant of proportionality
0:51If it's directly proportional, a fixed number comes with it: the constant of proportionality. The formula then has the form y is a times x. y is on the vertical axis, x on the horizontal one, and a is the constant of proportionality. For a spring, that is exactly the formula for the elastic force: F v is C times u.
1:14If you plot the elastic force against the extension, you get a straight line through the origin, and the constant of proportionality is C. You get it from the graph. Your turn. In a graph of the elastic force against the extension, a straight line runs through the origin and through the point eight point zero newtons at zero point two zero metres.
1:37How big is the spring constant?
C from the graph
1:43Pick a point on the line that lies exactly on a grid line, and as far from the origin as possible. Then you read it most precisely. Given: the elastic force is eight point zero newtons, and the extension zero point two zero metres. Asked: C in newtons per metre.
2:01Calculation: eight point zero is C times zero point two zero, so C is eight point zero divided by zero point two zero. Answer: the spring constant is forty newtons per metre.
Three things to remember
2:13Three things. Directly proportional means: twice as big gives twice as big, and the graph is a straight line through the origin. The formula has the form y is a times x, and a is the constant of proportionality. For a spring, that is C, and you find it with a point from the graph.
Two questions
2:33Two questions. One. A spring stretches four point zero centimetres at two point zero newtons, eight point zero at four point zero newtons, and fourteen centimetres at six point zero newtons. Is the extension directly proportional to the force?
2:47Two. Another spring stretches zero point zero five zero metres at three point zero newtons. How big is its spring constant?
The answers
3:01The first: up to four newtons it is, because twice as much force gives twice as much extension. But at six newtons it should have said twelve centimetres, and it became fourteen. So over the whole range it is not directly proportional: the spring has been stretched too far.
3:17The second: three point zero divided by zero point zero five zero is sixty newtons per metre.
Force meter
3:24Want to know how to measure a force, and which force meter to choose for it? Then watch the video about the force meter on my YouTube channel.
Frequently asked questions
When is a relationship directly proportional?
When one quantity becomes twice as big, the other also becomes twice as big. The graph is then a straight line through the origin, the point where both are zero.
What is a constant of proportionality?
The fixed number that comes with a directly proportional relationship. The formula has the form y = a · x, with y on the vertical axis, x on the horizontal axis and a the constant of proportionality. For a spring, that is the spring constant C in F = C · u.
How do you find the spring constant from a graph?
Pick a point on the line that lies exactly on a grid line and as far from the origin as possible, so you read it most precisely. Divide the elastic force by the extension: at 8.0 N and 0.20 m, C = 8.0 / 0.20 = 40 N/m.
Is a straight line always directly proportional?
No. The line must also go through the origin. A taxi with a four-euro starting fare and two euros per kilometre gives a straight line that starts at four euros: five kilometres costs fourteen euros, ten kilometres twenty-four, and that is not twice as much.
What is the spring constant if a spring stretches 0.050 m at 3.0 N?
C = F / u = 3.0 / 0.050 = 60 N/m.
Directly proportional: twice as big gives twice as big, and a straight line through the origin.
Physics with Thomas makes physics explainers for secondary school. No words on screen, so what you see works in any language.
In this series
Types of forces3:06
Calculating gravity3:15
Centre of gravity2:48
Force meter: range and precision3:11
Elastic force and spring constant3:41
Directly proportional relationship3:37
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.