Raindrop.mcd

11/2004

Raindrops keep fallin' on my head...

Objective:

In this lab we will consider the motion of a falling body with air resistance.

In class, we have studied the motion of a falling body. We were given the following data:

Time in seconds Distance fallen, in meters

and so forth

Here is a quick review of what we did in class: In Chapter 2,we computed the average speed during the first second, the first three seconds, the first five seconds, and the last two seconds, and noticed that the object was speeding up as it fell. We drew a scatter plot of distance vs. time:

and decided that this looks like the graph of a parabola s = ct². In Chapter 3 we looked at how to determine this constant c ...

Review of Section 3.3:

When we first considered the speed of a falling body, we considered only the force of gravity. The force of gravity is directed downwards, with a magnitude of mg, where m is the mass of the object and g is the (constant) acceleration due to gravity, approximately 9.8 meters, or 32.2 feet, per second per second

An important law of physics (Newton's Second Law) tells us that

Force = mass x acceleration .

So the force of gravity is equal to the mass of the falling object times the acceleration of the falling body.

We know what the force of gravity is; it's mg. That's the left-hand side of the equation.

For the right-hand side...The mass of the object is just m. The acceleration is the rate at which the velocity is changing; it is the derivative of velocity. So we have

force of gravity = mass of object x acceleration of object

or, dividing by m,

We would like to solve this differential equation for the velocity v. What kind of function has a constant for its derivative? In particular, what function of t has the constant g as its derivative?? Since we also assume that the velocity is 0 when we drop the object (at time 0), it is easy to guess that the solution is

NOW ... remember that velocity is the (instantaneous) rate at which distance is changing. In other words, velocity is the derivative of the distance function s(t) :

So s(t) is a function whose derivative is gt, where g is a constant. What kind of function has gt for its derivative? We assume that the object has fallen 0 meters (or feet, or whatever) at time 0, so the only solution of our differential-equation-with-initial-value problem

Does anyone recall what it was we were doing...?? Oh, yes, we were finding the constant c in the definition of the function s(t) = ct². We have just shown that

(a) this is an appropriate form of the function s(t); and

(b) the constant is g/2. If we are measuring distance in meters, g = 9.8. If we are measuring in feet, g = 32.2.

Air Resistance

In this lab, we investigate the corresponding problems that arise when we take into consideration the resisting force of the air through which the object falls. The usual physical assumption is that the resisting force is proportional to some power of the velocity, but the particular power (first, second, other) depends on the size of the object. In our case, we'll consider raindrops falling from a cloud 3,000 feet above the ground.

Exercise: If we ignore air resistance, as we did above, how long (in seconds) would it take a raindrop to fall from a height of 3,000 feet? How fast would it be traveling when it hit the ground? Give your answer in feet per second, and then change it to miles per hour. Show below how you computed your answer. (If you need the square root symbol, try View ... Toolbars ... Calculator.) Remember, when we are measuring distance in feet and time in seconds, g = 32.2)

A Mathcad trick to check your conversion:

Type in below (in math, not text) what your answer was, like

velocity := 700*ft/sec

Now we want to ask Mathcad to change this to miles per hour for us.

Type velocity = . You will see a small black box at the end of the answer. Enclose it in a small blue (half) box, and choose Insert then Unit...

Tell the computer that the Dimension is Velocity and then choose mph. (You may have to hit return to see the answer.)

Adding in air resistance

If a raindrop is small, say a drop of diameter 0.00025 feet (which is 0.003 inches, a size found in drizzle), we know that the resistive force is proportional to the first power of the velocity. In other words, the force due to air resistance is -kv. Here the minus sign indicates that the force is in the opposite direction to the velocity, i.e., upward rather than the positive downward direction.

So the drizzle drop has two forces acting on it -- gravity pulling it down, and air resistance dragging on it and "pulling" it up. It is still true that force = mass x acceleration, so we have

mass x acceleration = force of gravity + resistive force

We'll let a represent the quotient of the two constants k and m. When we add in our initial condition, we have a new differential-equation-with-initial-value problem:

Experimental evidence gives an approximate value of 52.6 for a.

We want to understand how, under these assumptions, the velocity varies as a function of time. We would like to graph the function, and maybe calculate a few values for particular times t. This differential equation is much more difficult to solve than the one above. We will take a numerical approach and use the computational abilities of Mathcad to find values of the velocity v at specific times.

We define the acceleration due to gravity

feet/sec²

and the constant a

We will calculate approximate values for the velocity at n equally spaced points in the interval from 0 to 0.2 seconds. In order to make it easier for you to change later, the value of n is assigned as a global variable at a convenient spot farther down in the worksheet. (That means that, no matter where you are in this worksheet, up above that definition or down below it, Mathcad knows the value of n.) The initial value of n is 20.

Since the length of the interval is 0.2 seconds, and we are dividing it into n sub-intervals, we have

The time points themselves, denoted t_i for i varying between 0 and n, are determined by the formula

The idea is to calculate approximations v₀, v₁, ..., v_n to the velocity at times t₀, t₁, ..., t_n. Well, we certainly know what v₀ is; the initial condition sets v₀ = 0. Fill this in on the line below:

We also know that the slope of the graph of v versus t at a particular time is given by the derivative there:

But at time 0, v = 0, so the slope at that point is just g. We also know that the average slope on the interval from time 0 to time 1 is

Therefore

Since v₁ is just supposed to be an approximation to the velocity at t₁, we can let it be determined by the formula

Solve this equation for v₁ and enter the answer below: You can type a D if you choose View ... Toolbars ... Greek. Remember to use * for multiplication!

Now we want to find v₂. Again, the slope at t₁ is determined by the differential equation

And, whatever value we choose for v₂, we want

So let's define v₂ so that that relationship is true.

Solve this equation for v₂ by hand, and then enter your result in the worksheet in the place provided below.

Continue this process to determine a value for v₃, and enter it on the worksheet.

It is clear that it will be tedious to enter definitions for all 20 values of v_i on the worksheet! To make matters worse, we plan to increase n to several hundred. At this point we need to remember the recursive definitions we made in earlier labs. Look carefully at your formulas for v₁, v₂, and v₃. Make sure you know what the pattern is. That means that, if we know the value of v_i, we can always find the next value, v_i+1. Write down the general recursive formula for v:

Before you move on, have your instructor or lab assistant check your answer.

This method of approximating the solution of an initial value problem by "bootstrapping" forward, using the differential equation and slope = rise/run, is called Euler's Method. You will probably recognize that we used this method, without the name, in the Limited Population Growth lab.

Once you have entered your formula and pressed Enter, all n values of v_n are computed. Now we are ready to plot velocity versus time:

This is the global definition of n. (Note the special equals sign. You can

find it in the Evaluation and Boolean pallette above.)

Change the value of n to 40 and see how the graph changes. Keep changing n, and create plots for increasing values of n between 20 and 100.

Eventually (but not for a couple of weeks), you will be writing up your results in a report.

For now, jot down your answers to the following questions:

1. Describe the sequence of plots and how they change as n increases.

2. Describe how the velocity of the falling object varies as time increases.

3. Especially notice that the velocity seems to approach a limiting value, called the terminal velocity. Use your calculated values of v_i to estimate the terminal velocity.

4. You will notice that the raindrop approaches the terminal velocity rather rapidly. Estimate the time it takes the drop to fall to the ground from 3,000 feet by assuming that the velocity is the constant terminal velocity the whole time. How does this answer compare to your answer to the Exercise above?

Print out a copy of today's lab and bring it to lab next week.