Acceleration is a simple idea – suppose a car increases its speed from 0 to 60 km.h^{-1} in 3 s. Its final speed is 60/(60 × 60) km.s^{-1}, so its average acceleration is 60/(60 × 60 × 3) = 0.006 km.s^{-2} (see post 16.13 for more explanation, and post 16.7 if you’re worried because the answer is 0.06 instead of 0.005556). However, if we want to calculate an acceleration that is not constant, and we want to calculate it at an instant in time (not just its average value), the calculation becomes more complicated. Then we need to use a mathematical technique called *differentiation* (post 17.4). So, a complete definition of acceleration involves this technique. If we want to use measurements of changing acceleration to calculate speeds or distance travelled, we need to use another mathematical technique called *integration*, this reverses the process of differentiation (post 17.19).

There is another complication – acceleration is a change in velocity which is like speed but includes the direction of motion (posts 16.2 and 17.4). So, an object moving around a circular path with a constant speed is accelerating, because its direction is changing (post 17.12).

As a result, acceleration can’t be represented by an ordinary number (a *scalar*) – it has direction as well as magnitude and so is represented by a *vector* (posts 16.50, 17.2 and 17.3).

Why didn’t I write that acceleration was defined by differentiation of a vector, when I first introduced the concept in post 16.13?

The reason is that it’s just too complicated to introduce the idea in this way. We only need the further sophistication, a vector defined by differentiation, when we want to solve more complicated problems than the average acceleration of a car travelling on a straight road. Most people don’t need to worry about vectors and differentiation to use the idea of acceleration. And it would be much more difficult to introduce ideas like vectors and differentiation before writing about velocity, acceleration and force because the explanations would be very abstract and so difficult to understand.

As a result, explaining science has been described as a process of “diminishing deception”. Usually we start by introducing a special case, that is easy to understand, and only make explanations more general when we need the added sophistication.

When we solve real problems, we can often ignore that velocity, acceleration and force are vectors. If motion occurs in a straight line, we can represent these quantities as ordinary numbers that are positive for motion in one direction and negative for motion in the opposite direction. I will continue to do this in future blog posts when it makes the explanation easier.

Complicated mathematics is used to make difficult concepts easier to understand and to solve difficult problems; it’s purpose is not to make scientific ideas difficult to understand.

*Related posts*

17.14 Confusion

16.41 Physics, chemistry and biology

16.18 What is life?

16.15 Science education