Calculus Online: Lab 4
Welcome to Lab 4 of Math 100 Sections 103, 104, 107 and 109.
Instructions
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Differential Equations
By now, you've probably seen the simple differential equation
in class. This lab aims to study differential equations in a bit
more detail. In particular, we want you to understand
 what a differential equation is,
 the role of an initial value and
 how differential equations can be used to make predictions.
Question 1: The Parachutist
You may remember that we looked at the position and
velocity of a ball which fell from rest on the first day of class.
Our study showed that the acceleration was constant. In some
situations, however, things aren't quite so simple: for instance, air
resistance is oftentimes an important force which works to oppose the
graviational force. This in turn will effect the acceleration causing
it to be no longer constant.
Shown below is the graph of the vertical velocity v
of an object thrown out of an airplane as a function of time
t. (Please make sure that "Question 1" is selected.)
Notice that the velocity starts to increase as the object falls faster
and faster due to gravity. However, air resistance causes the
velocity eventually to approach a constant called the terminal
velocity.
This graph is part of a familiar device, the Slope Meter shown to
the right of the graph, that we
used in Lab 2 to measure the slope of a secant line and the
derivative of a function. You can move the red dots on the velocity graph
so as to measure the slope of the secant line on the grid to the right.
When the two dots are very close together at some point, this slope is
approximately equal to the derivative of the function at that point.
Slope Meter
Data Plotter
Part a (2 marks)
Select "Question 1" on the Slope Meter: you will be
asked to measure the derivative v' of the function
v(t) at four or five points and then to record
v' as a function of v. For
instance, measure the derivative at the points where v = 0.5,
1.0, 1.5, .... (It may seem more natural to plot it as a
function of t, but please be patient and you will
see why we do it this way.)
To record your measurements, you will use the Data Plotter shown
just below the Slope Meter. Select "1 a" on the Data Plotter and you
will see a graph on which data points may be entered simply by
clicking the mouse where you would like a point. Data points may be
removed by clicking on an existing point.
Part b (2 marks)
Your data points should appear to lie on a line.
Select "1 b" on the Data Plotter. You will see two dots which
may be moved and a line which moves with them. Move the dots so
that the line passes through (or at least close to) your data points.
Let's think about what this is saying: we have a graphical
relationship between v' (plotted on the "y axis")
and v (plotted on the "x axis"). That graph is
just a straight line with a negative slope. But we know that a
straight line in the xy plane can be described by a very simple
expression of the form y = mx + b, or simply
y = (slope) x + intercept. This allows us to
conclude that our straight line relating v' to
v has the form:
where k and l are positive
constants which label the intercept and slope.
This is a differential equation since it describes the
function v by telling us about its
derivative. (You may want to compare this to the differential
equation
which describes y
by telling us about it derivative.)
Furthermore, the function which tells us the velocity of the falling
object is a solution of the differential equation: at every
point, the derivative v' and the value of the
function v are related by the differential equation
.
The following demonstration aims to make this more clear.
On the left, you are shown the graph of the velocity function again.
You can move a point around on the graph and watch how the tangent
line changes. On the right, you are shown a graph of the derivative
v' versus v;
that is,
You should notice
that the slope of the tangent line is represented by the height of the
dot on the right.
(There is no question associated with this demonstration.)
Notice that the differential equation predicts that the function
will continually increase until it reaches a velocity for which the
derivative is zero.
Question 2: Acceleration
Besides describing its solutions, a differential equation has a
great deal more information. For instance, remember that
acceleration is the derivative of velocity. That means that another
way to write our differential equation is
a = klv
Part a (1 mark)
Suppose that a falling object obeys the differential
equation you found in Question 1. What is the acceleration when it is
at rest? Select "Part a" below and enter your answer. (No units,
please, just a real number.)
Part b (1 mark)
Suppose that the velocity of an object obeys the
differential equation you found in Question 1. If its acceleration is
zero, what is its velocity? Select "Part b" above and enter your
answer. (No units, please, just a real number.)
Question 3: Satellites and Initial Values (1 mark)
Now we will repeat Question 1 only with a different
function. When you go back to the Slope Meter and select "Question
3", you will see the graph of the velocity of a satellite which is
returning to earth from space. Also select "Question 3" in the Data
Plotter (your data from Question 1 will still be shown so that we
might compare the experiments afterwards). As before, record the
derivative of this function v' as a function of
v at four points.
You should see that your data points in Question 3 lie on the same
line as the graph from Question 1. This is saying that this function
obeys the same relationship between the value of the function
v and its derivative v'. In other words,
it is another solution to the differential equation.
The functions in Question 1 and Question 3 both obey the same
relationship between v' and v
and so we have found two solutions to the same differential equation.
Let's think about why we could have two solutions to a differential
equation. The differential equation expresses how the velocity of a
falling object will evolve:
We may think of this as arising physically: the gravitational
force and wind resistance are acting in different directions and the
acceleration results from the net force.
However, the differential equation does not say anything about how
a falling object begins falling. In Question 1, we threw
an object out of an airplane. Once it was thrown, the differential
equation governed how the velocity evolvedin this case, it picks up
speed until it reaches a terminal velocity. In Question 3, a
satellite returning to earth originally has a large velocity.
However, wind resistance slows it down until it reaches its terminal
velocity.
The same differential equation produces two different solutions if
we start with different initial velocities. In fact, you
should not be too surprised that every inital velocity produces a
unique solution to the differential equation. The demonstration below
shows you on the left what the solution looks like when you
vary the initial velocity. On the right, you see the graph of the
differential equation with the initial velocity indicated with a dot.
(There is no question associated with this demonstration.)
Notice that the initial velocity v = 3 is
special: for this velocity, the solution is constant. You can see
this from the differential equation since the acceleration is zero for
this velocity. If the initial velocity is smaller than 3,
the acceleration is positive and so the velocity increases.
In fact, the differential equation says that the velocity will
continually increase until it is close to v = 3, which
is where the acceleration is zero. In the same way, when the
initial velocity is larger than 3, the acceleration is
negative so the velocity decreases and does so continually until it is
near the point where the acceleration is zero at v = 3.
Question 4: Population Dynamics
Differential equations tell us how quantities change and hence
they are useful for making predictions. For example, from the
differential equation for the acceleration of a falling object that we
have been studying, we could predict that there is a terminal
velocity: for small velocities, the acceleration is positive so the
velocity will increase until the acceleration is practically zero.
For large velocities, the acceleration is negative so the velocity
will decrease until the acceleration is practically zero.
Differential equations are equally useful in biology and we will now
discuss an example that is motivated by ecology. In ecology, it is
desirable to predict the long term behaviour of a population (for
example, to determine if the species is threatened by extinction.)
Shown below is a differential equation which describes how the
population of a species of fish evolves. The quantity y
measures the density of fish living in a particular habitat
and y' is the rate of change of the density.
Part a (2 marks)
Select "Part a" above and position the dot at an
initial value for which the fish will eventually become extinct.
Part b (1 mark)
Suppose the population density is initially around
2. Select "Part b" above and position the dot at the density of fish
that would result after a very long time.
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