Some examples of calculations outside the A-level syllabus, just for fun, to show how mathematics can actually be used.
The wind-powered car (using work&power from M2).
Sailing boats can "reach" across the wind, often with a higher speed than the wind itself. Can one, though, go straight downwind faster than the wind?
Can one go straight into the wind?
Try to calculate the maximum possible speed….here is my solution.
If you are interested in aerofoils, see the NASA FoilSim applet and Martin Hepperle's JavaProp.
To understand the difference between a propeller (for a downstream wind car) and a wind turbine (for an upstream wind car), see these aerofoils.
What is the colourful wavy surface?
This is a simulation of an 8" thick concrete floor. A huge Diesel engine sits in the middle; every time a cylinder fires, the torque reaction gives the floor a twist.
At low frequencies (left end), the whole floor pants up and down on each side.
At high frequencies (right end), you just get a little wiggle in the middle.
Generally speaking:
- an equation gives you a number
- a differential equation gives you a line
- a partial differential equation gives you a colourful surface.
If you do engineering, mathematics or physics at university you will find out about all this. Fun!