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by George E. Hrabovsky, President of MAST

News from MAST

Hello again! I hope everyone had a nice Thanksgiving. I know I did; good friends, good food, and several good movies.

Allow me to congratulate David Weiller on his web site. It is awesome! An excellent example of what the Internet can be used for when someone applies themselves. My hat's off to you, sir!

I would also like to make a comment about how one becomes good at mathematics. I can show you interesting tricks and principles until the clock runs out and it will not do you any good. Science in general, and theory in particular, is not a spectator sport! You have to get in there and play the game. When we do something successfully then we have learned a little (the technique works, we have calculated something of interest, we have made an interesting prediction, etc.) When we get something wrong then we learn a lot! We struggle for the correct answer, we struggle to find out where we went wrong, we struggle to understand what happened! By failing we launch an entirely new inquiry into our basic understanding of the problem at hand. So, we must learn to fail and fail often.

This week I will conclude our study of the geometry of motion.

The Inverse Problem

So far we have learned how to measure the position of an object, and to find its displacement, velocity, and acceleration when moving. What happens if we do not know its position in advance? What happens if we see an object moving and all we know is its acceleration. Take, for instance, an object dropped from rest (in other words no initial velocity) and from a height . When we drop the object it falls, it doesn't just hang there in the air. Since it begins with no velocity and starts to fall with some velocity that depends on time , we can assume that an acceleration is at work. What acceleration would it be? Since it is falling towards the ground it must be the acceleration due to gravity on the surface of the Earth. This has a special symbol in physics, . So, we begin with knowing only that we are dropping the object with acceleration from height . Now what?

Recall that acceleration is the time derivative of velocity,

We can reverse the order of this expression,

Recall from the previous column ("Adding it All Together") that there is an operation that is the exact opposite of differentiation, it is integration. To integrate this expression we need to separate the variables, that is put the and all related factors on one side of the equation and the and all its related factors on the other side. We do this, in this case, by simply multiplying each side by ,

We can now place the integral signs (we have to integrate both sides of the equation, since anything we do to an equation must either leave the equation unchanged or must be done to each factor in the equation),

Looking at the left-hand side of the equation we are integrating with respect to . Since there is no value of listed, it is a 1 (since ). To find the integral of 1 with respect to we ask the question, "What derivative of will leave us with the answer 1?" The answer to that question is (in its most general form),

Where is some constant. Recall that integration is adding up all of tiny slices below a curve. We can take the integral of a function from some starting point, say and stop at some ending point . A general integral would then look like this,

where is the derivative of , and , and . Using our integral,

The cancels out and we have,

What about the right-hand side of the equation? What is the integral of ? Recall from above that the acceleration is actually . This is a constant, in SI units it has the value . So we have,

Since the integral of a constant is the constant times the integral,

From our discussion above we have to find the derivative that gives us 1 when differentiated with respect to time. This is,

If we integrate from to some later time, we run into a problem; how can we tell what we are using? We adopt a place-holder called a dummy variable in the integral, and integrate that for 0 to . We will use the symbol for the dummy variable (its traditional).

We can now put the two halves of the equation back together again.

We can easily solve this for ,

In our case , so we have,

By a similar method we can arrive at the general formula for position at any time, ,

Or, in our case,

Theory Challenge Answer for Last Week's Column

Well, I just did it as the main body of the article this week.

Theory Challenge

Fill in the details of how to get the position with respect to time.

Books That I Like

Every good calculus-based physics text will have this in it. Here are some that I like:

Michael Mansfield, Colm O'Sullivan, (1998), Understanding Physics, John Wiley & Sons, in association with Praxis. This is a very neat book that rapidly develops the background necessary for theoretical work.

Hugh Young, Roger A. Freedman, (2000), Sears and Zemansky's University Physics with Modern Physics, 10th Ed., Addison-Wesley. This is an immensely good book that is also immensely expensive (I think I paid more than $100 for the damn thing)! It is a very good book, including covering common conceptual problems that students have.

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Converted by Mathematica      November 29, 2001