What is a Rate of Change?

An introduction to the topic of motion.

If you understand one thing before you start this module it should be what a rate of change is.


I could rephrase that, if you don't understand what a rate of change is, you are going to be lost throughout this module.  A rate of change is anything per anything else, most usually a change is something per unit of time.


I'll try and give you a good everyday example, well I guess pay would be a good one.  Jobs tend to pay per hour, we would call this your rate of pay.  The minimum wage is currently £7.20 per hour for those over 25 in the UK.  So if you wanted to work out total pay, you'd do rate multiplied by number of hours worked.  And if you knew how much you'd been paid and wanted to check your rate then you'd do a calculation; total pay divided by hours worked, and you'd calculate your rate of pay in pounds per hour.  And if for some reason you wanted to work out how much time you'd worked last week you'd look at your total pay and divide that by your rate.


This is pretty simple numeracy stuff, but working with rates in Physics is no more complicated than that.  It's just not something that you are necessarily used to thinking about.  And I think some students can get confused, especially when it comes to unit conversion!  We wouldn't every really need to see our pay in pennies per second but just as an example, here you go!


And remember, we are really talking about motion, I'm just introducing the general idea of a rate of change.


An example unit conversion:


The current minimum wage is £7.20 per hour, what is this same rate in pence per second?




Hopefully that made sense.  Speed is just a rate of change of distance, i.e. it's distance divided by time, or maybe you prefer to visualise that as one of the units of speed, metres per second (m/s), or miles per hour.  This really is as simple as working out rates of pay and total pay.  But instead of money the thing which is changing is position!


I'll show you using the equation for speed, I'm going to calculate the same speed in two different units.  This is worth following carefully, and maybe having a go at a different one yourself to check you can do it, because I really think this can be a block for some students learning about motion.  If you don't get this right it's something that will make it all a lot harder than it needs to be later in this module.



distance travelled (m) = speed (m/s) × time (s)


(I know it's weird we use s for distance, it comes from the Latin stem spatium, like as in space!  And v is also from a Latin stem and means velocity, which is the vector form of speed.)

Worked example:


What is the speed of a car cruising through a set of road works if it travels 25km in half an hour?


It's really important that you take on board that these are the same speed but they are just expressed in different units.  And remember it is always easiest to convert units as early as possible.


So being used to working with rates, really helps you understand these tricky maths bits that are coming up.  You need to be resilient and not get put off when there is a lot of numbers and algebra to deal with.  Work through these questions methodically and you'll be fine.


Other rates in motion and forces you'll come across really soon are:  Acceleration is a rate of change of speed, it's a change in speed divided by time, or a metre per second per second (m/s2).

A force is a rate of change of momentum and a mass is best defined as a force per acceleration, or a measure of an object's inertia!  (Don't worry too much about these ones just yet.)


And it is very true to say that rates are best shown on graphs, because a gradient is a very visual way of showing a large rate of change or a small rate of change.


Get all that?  Try these questions.

Then go on to the motion graphs section, or have a look at the difference between scalars and vectors.



1.  Calculate the speed of a runner in metres per second (m/s) who completes a 400m race in one minute.


2.  Calculate the speed of a train which completes a 400km journey in 2 hours.  Give you answer in km per hour and in metres per second.


3.  How far have I gone if I walk at a steady speed of 2m/s for a quarter of an hour?