The physicist's favourite question
Physicists rarely start by asking “what is the answer?”. They ask: if I double this, what happens to that? Double the speed of a car: does the stopping distance double? Double your distance from a lamp: is it half as bright? The answer is almost always one of four simple patterns, and knowing which one lets you predict results with no calculator at all.
A graph of y against x for one of four laws: y proportional to x, to x squared, to one over x, or to one over x squared. A grey point marks x = 1 and a blue point marks x = k, set by the slider. The readout says how many times bigger or smaller y became.
What happens to y when you scale x?
InteractiveThe law
The math, with your numbers
Example: distance at a steady speed, d = vt. Only the ratio matters: you never needed the constant in front.
Try this
- →Double x under each law. Which one quarters y?
- →With y ∝ x², what k makes y nine times bigger?
- →Set k = 0.5 (halving x). Which law makes y four times bigger?
Direct proportion: double means double
Two quantities are directly proportionaly = kx: when x is multiplied by some number, y is multiplied by the same number. The graph is a straight line through the origin. when multiplying one multiplies the other by the same amount. Walk at a steady speed for twice as long and you go twice as far. The symbol reads “is proportional to”:
The number (the constant of proportionality) is whatever makes the units work: in it's the speed. On a graph, direct proportion is a straight line that goes through the origin, and the ratio is the same at every point.
Square proportion: double means four times
When , doubling x multiplies y by , and tripling it by . The most important example in physics is kinetic energy, the energy of motion:
Say it out loud: “energy equals one half times mass times speed squared.”
Kinetic energy· measured in joules (J)
The energy something has because it's moving.
Mass· measured in kg
How much stuff is moving.
Speed· measured in m/s
How fast. It's squared, so it matters a lot.
Inverse proportion: double means half
When , the two move in opposite directions: double x and y halves. Drive a fixed trip twice as fast and it takes half the time. Here the product stays constant instead of the ratio: .
Its stronger cousin is the inverse-square lawy ∝ 1/x²: double x and y drops to a quarter; triple it and y drops to a ninth., : double the distance and y drops to a quarter. Gravity, electric forces, the brightness of a lamp and the loudness of a speaker all follow it, because whatever spreads out from a point is shared over a sphere whose area grows as the distance squared.
Make a prediction first
You stand 2 m from a lamp and move to 6 m. How bright does it look now?
Think about it, then click to reveal the answer.
Make a prediction first
You stand 2 m from a lamp and move to 6 m. How bright does it look now?
Think about it, then click to reveal the answer.
You tripled the distance. Brightness follows an inverse-square law, so it drops by : one ninth as bright. In the simulation: pick and set k = 3.
The ratio trick
All four patterns fit one rule. If (with n = 1, 2, −1 or −2), then
The constant cancels out, so you never need it. That's why you can answer “what happens to the force on a satellite if it orbits twice as far out?” without knowing the satellite's mass or Newton's constant.
Worked example: a car's kinetic energy
Step 1 of 4- 1Kinetic energy goes with the speed squared, so n = 2.
Wrap-up
Key ideas to remember
- Direct (): double x, double y. A straight line through the origin; is constant.
- Square (): double x, four times y. Kinetic energy and braking distance.
- Inverse (): double x, half y; is constant. Inverse square: double x, a quarter of y.
- Ratio trick: . The constant cancels, so you don't need it.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.