Models and Exponentials – A2 Physics Revision

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 = emx 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 = emx 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 = emx 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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About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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