Modelling

- Modelling is creating an artificial world which can be used to make predictions about a system
- Models are simplified and idealised, as they usually don’t take into account all of the smallest factors
- An example of modelling is the use of exponential equations to describe things like radioactive decay and capacitor charging / discharging
- (Further examples are the Boltzmann Factor and the energy loss in damped oscillators)

Graph Shape

- The rate of change is proportional to the quantity present
- If N is the quantity, then ΔN/Δt α N
- Exponential growth gets faster and faster because the rate of change increases as the quantity increases
- Exponential decay gets slower and slower because the rate of change decreases as the quantity decreases
- The graph of y = e
^{mx}shows exponential growth - The graph of y = e
^{-mx}shows exponential decay

Natural Log Graphs

- To get an exponential relationship to form a straight line, you can plot the natural log of the y-axis variable
- If you take logs of both sides of y = e
^{mx}you get ln y = mx, so plotting ln y against x will produce a straight line with the gradient m - Plotting a natural log graph to see whether it makes a straight line is one way to test whether the relationship between two variables is exponential
- Exponential growth should produce a natural log graph with a positive gradient
- Exponential decay should produce a natural log graph with a negative gradient
- The gradient is the variable x is multiplied by in the original equation (so the gradient of the natural log graph of y = e
^{mx}is x, and y = e-^{mx}‘s natural log graph would have a gradient of -m)

Constant Ratio Test

- Another method of testing whether a relationship is exponential is to perform a constant ratio test on the set of data
- A relationship is exponential if there is always the same ratio between consecutive y values (at equal x intervals)
- I’m not sure how clear I’ve made that. If you can tolerate an example, have a look at the two below

Successful Constant Ratio Example

- Measurements of variable y have been taken at equal intervals of variable x
- It’s important that x increases by the same amount each time – in this case, by 5
- Every y value (except the first) has been divided by the y value before it
- The result has always been roughly 1.4
- This ratio is fairly constant, so we can conclude that the relationship between x and y is exponential
- (If x and y were plotted, the result would be an exponential growth curve)

x | y | y divided by previous y (3 d.p.) |

10 | 470 | No Previous y available |

15 | 679 | 679 / 470 = 1.445 |

20 | 977 | 977 / 679 = 1.438 |

25 | 1408 | 1408 / 977 = 1.441 |

30 | 2025 | 2025 / 1408 = 1.438 |

Unsuccessful Constant Ratio Example

- Here’s one that fails…
- The ratio varies a lot more

x | y | y divided by previous y (3 d.p.) |

10 | 700 | N/A |

15 | 1575 | 2.250 |

20 | 2800 | 1.778 |

25 | 4375 | 1.563 |

30 | 6300 | 1.440 |

Data, Formulae and Relationships Booklet

- There aren’t actually any equations to learn in this post

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