Core Pure - Modelling With Differential Equations

Question 1

What different motions should you be able to model?

Answer 1

Terminal velocity
Mixing fluids
SHM

Question 2

How can you do terminal velocity questions?

Answer 2

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

Question 3

How do you do mixing fluids questions?

Answer 3

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

Question 4

What is SHM?
In words

Answer 4

SHM - Simple Harmonic motion is when the acceleration is always directed towards the origin an is proportional in magnitude to the displacement from O

Question 5

What is SHM?
In Math

Answer 5

Question 6

What is SHM in a graph?

Answer 6

It is at A when alpha=pi/2

Question 7

acceleration is also equal to:

Answer 7

Question 8

What is the maximum value of V in SHM?

Answer 8

Question 9

How do you calculate the Time period of 1 oscillation of SHM?

Answer 9

Question 10

What are the different types of SHM?

Answer 10

Normal
Damped
Forced

Question 11

What changes in damped SHM?

Answer 11

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

Question 12

What are the different types of damping?

Answer 12

Heavy
Critical
Light

Question 13

When does heavy damping occur?

Answer 13

d>0

Question 14

When does critical damping occur?

Answer 14

d=0

Question 15

When does light damping occur?

Answer 15

d<0

Question 16

What is heavy damping also know as?

Answer 16

Exponential decay

Question 17

What is critical damping also know as?

Answer 17

Critical decay

Question 18

What is exponential decay?

Answer 18

It goes to 0 very quickly

Question 19

What is light damping also known as?
Reduce it and explain?

Answer 19

Decaying SHM

Question 20

What does the graph of heavy damping look like?

Answer 20

Question 21

What does the graph of critical damping look like?

Answer 21

Question 22

What does the graph of light damping look like?

Answer 22

Question 23

Describe the graph of light damping?

Answer 23

It oscillates with a reducing amplitude (decreasing exponentially over time)

Question 24

How can you derive damped SHM?

Answer 24

F=ma

Just know this:

Question 25

What will be the difference with forced SHM?

Answer 25

The equation will be non homogenous

Question 26

What are coupled first order differential equation’s?

Answer 26

These are differential equations that include 2 dependent variables (usually with respect to time)

Question 27

What is usually the case with predator prey models?

Answer 27

Question 28

When is population growth stable…?

Answer 28

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

Question 29

What is the general method for solving coupled first order differential equations?

Answer 29

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