User:Addemf/sandbox/Who Invented Calculus?/Free Fall

Gravity
Imagine that you are a scientist in ancient times, trying to understand how gravity "works".

At least to get started, you might want to remove all other forces involved, and ask the following question.

Suppose that I hold a rock out in front of me and release it into free fall. That is to say, assume that the only force acting on the rock is gravity.

How does the rock fall?

In trying to answer the question above, we will find that the mathematics that we may know before learning calculus, is inadequate to answer the question. Therefore, we will invent calculus in order to answer it. In this narrow sense, one could say that all of calculus derives from this one question.

In particular, when we ask "how does it fall?" we mean something like,

More precisely, what we want is the position function for the rock. That is to say, we would like to know the position of the rock at each moment of time.

The Concept
For the rest of this lesson we will focus on the general concept of a position function. In future lessons we will return to our particular interest in the position function of an object in free fall.

Consider some arbitrary function, which we will interpret as a position function.
 * $$x(t) = t^2-t-1$$

Here t is time, and x is the position of some object. You may think of the object as a particle, or the center of mass of a person or car, or anything else that you like. We assume time is measured in seconds and position in meters.

The point of having a position function, is that it is a completely precise way of expressing where an object is, given any moment in time.

1. Find the position of the particle described by $$x(t)$$ above, at the times $$t=-1,0,1$$.

Note that negative times are perfectly sensible, since we usually use $$t=0$$ to mean something like "the current moment" or "the initial time". But there is always time before the current moment, represented by negative time.

2. Find the time when the position is $$x=1$$. If there are multiple values of time, then given the solution set.

Diagramming a Position Function
The diagram in the thumbnail to the right shows some arbitrary position function. That is to say, it is some function which we are meant to interpret as the position of some object.

It contains two different representations of the same function. At the top is a two-dimensional graph of x against t. Below that is a graph of only x, but we still see an animation of how x changes as time progresses.

Some of these choices are arbitrary. For example, here we are diagramming x against t, where x is usually understood to be horizontal motion.

But we could alternately diagram vertical motion. You can see, at the top of this page, a diagram with the axis y running down-up only. The symbol y is usually used for vertical motion. Since that diagram shows free fall, which is vertical motion, it is diagrammed accordingly.

Look back at the animated graphs to the upper-right of this text.

In the two-dimensional plot, the dot is above the t axis whenever the dot in the one-dimensional plot is to the                     of 0 there.