Core Pure - Modelling With Differential Equations Flashcards
What different motions should you be able to model?
Terminal velocity
Mixing fluids
SHM
How can you do terminal velocity questions?
You want to find the lim(v) as t tends to infinity
Use F=ma
Solve the differential equation
Re arrange for v
Find the limit
How do you do mixing fluids questions?
Draw a diagram
After t hours (work out the total volume) (initial + change(t))
Work out the sulphate in (a constant)
Work out he sulphate out (as it mixes instantaneously it will be a proportion of the total volume (out(x))/total volume )
dx/dt=In - Out
What is SHM?
In words
SHM - Simple Harmonic motion is when the acceleration is always directed towards the origin an is proportional in magnitude to the displacement from O
What is SHM?
In Math
What is SHM in a graph?
It is at A when alpha=pi/2
acceleration is also equal to:
What is the maximum value of V in SHM?
How do you calculate the Time period of 1 oscillation of SHM?
What are the different types of SHM?
Normal
Damped
Forced
What changes in damped SHM?
There is a damping force acting against the motion if the particle. The force is proportional to V.
It can be seen as a dy/dx term in the equation
What are the different types of damping?
Heavy
Critical
Light
When does heavy damping occur?
d>0
When does critical damping occur?
d=0
When does light damping occur?
d<0
What is heavy damping also know as?
Exponential decay
What is critical damping also know as?
Critical decay
What is exponential decay?
It goes to 0 very quickly
What is light damping also known as?
Reduce it and explain?
Decaying SHM
What does the graph of heavy damping look like?
What does the graph of critical damping look like?
What does the graph of light damping look like?
Describe the graph of light damping?
It oscillates with a reducing amplitude (decreasing exponentially over time)
How can you derive damped SHM?
F=ma
Just know this:
What will be the difference with forced SHM?
The equation will be non homogenous
What are coupled first order differential equation’s?
These are differential equations that include 2 dependent variables (usually with respect to time)
What is usually the case with predator prey models?
When is population growth stable…?
Population growth is stable when the rate of growth is 0
So if the answer tends to not 0 population growth won’t be sustainable
What is the general method for solving coupled first order differential equations?
Differentiate the first equation with respects to t to get a second order derivative (in terms of dx/dt and dy/dt)
Substitute in the value for dy/dt from the second equation
From the first (original non differentiated equation), make y the subject
Substitute y into the ongoing equation
Simplify to get an expression in terms of x(t)
Solve it
Differentiate the answer
Substitute x(t) and x’(t)
Into the original equation and rearrange (make sure to pull out as many factors as possible)
Rearrange to make y the subject
State the answers