Calculus Online: Lab 4

Welcome to Lab 4 of Math 100 Sections 103, 104, 107 and 109.


Instructions

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  3. All submissions must be received by 11:59 pm on 9 November 1998.

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Differential Equations

By now, you've probably seen the simple differential equation

\[ 
\frac{dy}{dt} = ky 
 \]

in class. This lab aims to study differential equations in a bit more detail. In particular, we want you to understand

  1. what a differential equation is,
  2. the role of an initial value and
  3. 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:

\[ 
v' = k - lv 
 \]

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 $  \frac{dy}{dt} = ky  $ 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 $  v^\prime = k - lv.  $ .

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, $  v^\prime = k - lv.  $ 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 = k-lv

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:

\[ 
a = v^\prime = k - lv 
 \]

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 evolved--in 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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