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Skip to 0 minutes and 13 seconds All right so let’s talk about inverse relations. These are important new relations to add onto our linear or direct proportions and our quadratic relations. So the basic idea is we have one quantity which is inversely proportional to another. It means that y equals a over x for some fixed constant a. In other words, these two quantities x and y are related in such a way that their product is always fixed. A very good example is what happens when we are going from one place to another and observing how long it takes, how far we’re going, how fast we’re going.

Skip to 0 minutes and 51 seconds The three quantities– velocity, distance, and time, of course– are related by this familiar equation which we can rewrite as velocity times time equals distance. That’s telling us that if the distance is fixed– let’s say you’re going from here to here– then the relationship between the velocity and the time is an inverse relation. You double the velocity, then you half the time. You double the time, same as halving the velocity. If we wanted to graph the relationship between v and t, we would get a graph that looks like the 1 over x graph, except that the constant would not be necessarily 1.

Skip to 1 minute and 34 seconds So let’s say with a velocity v1 it takes you time t1, a velocity v2, it takes you time t2– then this is the kind of relation that we have. If v2 is twice v1, then t2 will be 1/2 t1. So one very important example of an inverse proportionality is in the ideal gas law. This is a fundamental formula in chemistry where we’re interested in what happens to a gas which is contained inside some region. So for example, here’s a gas contained in a cylinder, and have some kind of piston that’s able to compress the gas, so to change its volume.

Skip to 2 minutes and 14 seconds So a natural question is, what happens to the pressure that that gas is exerting when its volume is being, say, decreased? And the law is that that pv equals nrt. p is pressure, v is volume. n is the amount of gas. r is a constant, and t is actually the temperature. But if we’re considering a situation where the temperature is fixed, then we can think of this right hand side as being a constant. So it’s really an equation of the form p times v is some fixed constant k. And that’s an example of an inverse proportionality. If we increase the volume, then the pressure decreases. If we decrease the volume, the pressure increases in a very predictable way.

Skip to 3 minutes and 5 seconds So if you multiply the pressure by 3, then correspondingly, you have to divide the volume by 3, and so on. Another important example is in the theory of electrical circuits a fundamental law called Ohm’s law, which tells us about the relationship between the voltage, v, the current, i, and the resistance r in a basic circuit. So if we have, say, a battery here that’s applying a constant fixed voltage to the circuit, and here’s some kind of resistor– perhaps like a toaster– then a certain amount of current is going to go through that circuit.

Skip to 3 minutes and 47 seconds So Ohm’s law is that v equals ir, or we can rewrite it as ir equals v, telling us that we have an inverse proportionality between the current i and the resistance r assuming that the voltage is fixed. So for example, if you have a battery of a fixed size, or if you’re, say, plugging into a household appliance with a fixed voltage, the voltage is fixed, and then there’s this inverse proportionality, an inverse relationship between the current and the resistance. So we get a graph that looks like this again. So in previous weeks, we’ve looked at direct proportionalities where the two quantities are on opposite sides of an equation. So here we’re doing something which is different.

Skip to 4 minutes and 32 seconds So we’re expanding our toolbox. It’s enlarging our point of view, encompassing more things that we can study and analyse still in a relatively simple algebraic way.

Inverse relations from travel, gases and electricity

In this video we give an overview of different examples of inverse relations associated with physics: velocity, Boyle’s law and Ohm’s law.

Two quantities have an inverse relation if their product is constant. Another way of saying this is that one quantity is the inverse, or reciprocal, of the other. This is a relational way of describing the function \(\normalsize y=1/x\).

Examples include the relation between velocity and time if we need to travel a fixed distance, between pressure and volume in Boyle’s law (and its generalisation the Ideal Gas Law), and between current and resistance in a battery driven circuit, a key result in physics called Ohm’s law.

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This video is from the free online course:

Maths for Humans: Inverse Relations and Power Laws

UNSW Sydney