Before you read this, it would be useful to read post 16.8.
Post 16.8 is a story about a biologist who investigated the rabbit population on a farm. He counted the number of rabbits every year for 5 years. When he plotted a graph of the number of rabbits against time, he got a straight line. He used this straight line to predict the number of rabbits in the future. The straight line provides a model for the rabbit population. In science, a model doesn’t usually mean a physical object – it’s simply something we use to explain or predict what happens around us.
Unfortunately, the predictions of the model were incorrect. The reason was that, a few years later, foxes came to the farm and started to eat the rabbits. The biologist’s model did not allow for this possibility and so it didn’t work.
I recently asked a group of students what they would have done if they had been the biologist – most of them decided that they would need to develop a more complicated model that allowed for all the factors that could influence the rabbit population.
This approach won’t work! Just think of all the factors that could influence the rabbit population – the population of other predators (wolves, snakes, hawks and other animals, as well as foxes), the prevalence of diseases (like myxomatosis), the population of alternative food for the predators (chickens, mice and lots of others), the supply of food for the rabbits, the occurrence of poisonous plants mixed in with the food plants, the effect of the weather on the growth of food plants and poisonous plants – you can never be sure that you’ve thought of all of them.
It gets worse. To develop such a model, you need reliable numbers to describe how all these factors interact with each other. However, when we measure these numbers, there will be some uncertainty associated with them (see posts 16.24 and 16.26). All these uncertainties will combine, so that the model makes imprecise predictions.
We need to keep our models simple. Let’s suppose we want to predict the speed at which an object of mass m, falling from a height h, hits the ground. The simplest model for this process is that the falling object converts gravitational potential energy, mgh, where g is the strength of the gravitational field (post 16.16), into kinetic energy ½mv2, where v is the speed at which the object hits the ground (see post 16.21). If none of the potential energy is used in other processes
mgh = ½mv2.
If we divide both sides of this equation by m and then multiply both sides by 2, we get the results that
v2 = 2gh.
So, the speed at which the object hits the ground is given by
v = √(2gh).
Here the “square root” sign (√) means the number we have to multiply by itself to get the number in the brackets; for example √(25) = 5 because 5 × 5 = 25.
This model for the speed of a falling object is based on the principle of conservation of mechanical energy (post 16.21). Are we sure that all the mechanical energy is conserved by a falling object? What is the effect of air resistance (drag) on the falling object? Drag is important in design of cars to improve efficiency and reduce fuel consumption. It depends on the shape and dimensions of the object and the viscosity of air. We are familiar with liquids like lubricating oils and treacle as having viscosity – air has viscosity too. Overcoming drag is similar to overcoming friction when two solids are in contact – the difference is that here a solid object is in contact with air. Temperature affects the viscosity of air; it also affects the dimensions of the falling object (post 16.35). Temperature differences in the air will cause convection currents that could affect the motion of objects.
If we want to calculate the speed of a falling feather, we might have to think about some of these factors; if we want to calculate the speed of a falling lump of metal, we ignore drag and all the associated factors. We know that these factors exist but that they are negligible. We use the simplest model that works – a model based on the principle of conservation of mechanical energy.
So, what did the biologist, in the story of post 16.8, do wrong? He ignored a factor, the possibility that foxes would come to the farm, without thinking about whether it could be safely neglected.
What should he have done? He should have performed an experiment to test his straight-line model (post 16.3) before he made any predictions. An experiment can’t prove that a model is correct but it can show if it doesn’t work (see post 16.3). Testing is an essential step in developing any scientific model; we’ll return to this subject in the next post.
16.41 Physics, chemistry and biology
16.36 Good and bad
16.32 Faith in science
16.28 Significant differences
16.26 Normal distribution
16.22 Science can’t explain everything
16.10 Expensive cars and health
16.3 Scientific proof
16.2 Scientific laws
16.1 Drug safety
16.43 Test before you predict
18.6 The pendulum: a simple harmonic oscillator
18.25 An ideal gas
22.17 Coupled differential equations