User:Addemf/sandbox/Who Invented Calculus?/Position Velocity Acceleration

In the previous lesson we learned about the position function. Recall that this is a mathematical expression of, "You tell me the time, I'll tell you the position."

In this lesson we will try to understand how the position function is determined by other functions, ones which we may understand better.

Constant Velocity
We often try to reason about a position function by first having some idea about how the object moves. For example, if an object moves at a constant rate, then we can use this information to say something about the position function.

Suppose that an object has position function $$x(t)$$ and starts at position $$x=2$$. That is to say, $$x(0)=2$$. (Note that the "start" or "initial" position is the position that the object has when $$t=0$$. The time, $$t=0$$ is also called the "start" or "initial" time.)

Further, suppose that the object moves toward the right at a rate of 3 meters per second.

Write the equation of the position function and plot it.

Suppose that an object has a constant velocity of -5 m/s. Initially it was located at position $$x=10$$ and then 4 seconds elapse. What is its final position?

In the above exercise, we assumed a constant velocity throughout the motion of the object.

Could the answer to our question above, be that objects move with constant velocity?

Since we are being scientific about this question, let us formally state a hypothesis for this idea.

Suppose that you hold an object above the ground and release it into free fall. Then the object will fall at a constant rate.

Probably this hypothesis is intuitively false. It is instinctively clear that objects fall faster and faster, so the rate at which they fall should not be constant.

Still, since we are being scientific, let us formally disprove this hypothesis. It will be good exercise to develop principles about hypothesis proof and disproof in this easy setting. That way when hypotheses become hard to prove or disprove, we will have developed some skills for dealing with them.

Design an experiment to either prove or disprove The Constant Velocity Free Fall Hypothesis.

Non-constant Velocity
Since we intuitively know that velocity increases as an object free falls, then we should think of velocity as a function of time. That is to say, we should consider the value of velocity, at each moment in time.

This idea is in perfect analogy with our idea of a position function. Just as a position function represents an answer to "Tell me the time, I'll tell you the position", likewise the velocity function represents an answer to "Tell me the time, I'll tell you the velocity."

We often use the symbol v for velocity, and therefore write $$v(t)$$ to express velocity as a function of time.

For example we might have $$v(t) = t^2$$. This would imply, for one thing, that $$v(0)=0$$. Translating this into its physical interpretation, this means "initially the object is at rest". We say that something is at rest when its velocity is zero.

For another thing, this velocity function implies that $$v(1)=1$$, which is to say, after one second has passed the velocity has increased to 1 m/s.

Note that the units of velocity are meters per second, which we abbreviate with the symbol "m/s".

Graph the velocity function $$v(t)=t^2$$, putting the axis v vertically, and t horizontally.

Based on the graph that you draw, argue that for any time after $$t=0$$ (i.e. for any time $$0<t$$), the position of the object must be somewhere toward the right of its starting place.

Change and △
Much of our study is about how quantities, like position and velocity, change over time. Therefore it is worth establishing some notation and vocabulary for ideas related to change in general.

Note that, in the definition above, a and b are values of time if f is a position function. Also, both displacement and distance are measured in meters.

Let $$f(x) = x^2-5x+2$$.

1. Find the change in f from 2 to 3.

2. Find $$\Delta_2^3 f$$.

Assume that f is a position function for the following parts.

3. Find the displacement of f from 2 to 3.

4. Find the distance traveled by f from 2 to 3.

5. In general, a negative value of displacement means that the object traveled.

Compute each of the following.

1. $$\Delta_{x=0}^{x=1} x^2$$

2. $$\Delta_1^2 \ln x$$

3. $$\Delta_{-2}^{-1} t^2$$

Let $$f(z) = z^3$$

4. $$\Delta_{z=0}^{z=5} f$$

5. $$\Delta_a^b t^2$$

6. $$\Delta_3^7 t$$

7. $$\Delta_a^b t$$

If the velocity of an object is constant, then we can write a simple expression for velocity using this change notation. It essentially says that velocity is the "change in position over change in time".

Note that, in the above expression, it was useful to make the variable for time explicit in the numerator. That is why I've written $$\Delta_{t=a}^{t=b}x$$ instead of just $$\Delta_a^bx$$. This is because x is a symbol for a function of t, so when writing the $$\Delta$$ notation, it becomes unclear which is the variable we should substitute values for. The explicit notation removes this ambiguity.

However, for the denominator, there isn't as much of a chance for misunderstanding. t is a basic variable and not a function of anything else.

Suppose that an object moves at a constant rate with position function x. Also suppose $$x(-2)=1$$ and $$x(2) = 4$$.

Find the constant velocity of the object.

Velocity is only defined above for an object moving at a constant rate. We can extend the notion to objects with a non-constant rate. But then we have to be very aware of the fact that our calculation of the velocity is only an approximation ― or, more precisely, it is an average taken over the time interval.

Notice that the notation for average velocity includes a "bar" over the symbol v. Throughout much of mathematics, a bar over a symbol indicates taking an average.

Let $$x(t) = \sin t$$ be a position function.

1. Find its average velocity from 0 to π.

2. Find $$\overline v_{\pi/2}^{3\pi/2}$$.

Velocity and Position Graphs
The diagram to the right simultaneously shows a position function, x, and a velocity function v, super-imposed onto the same graph.

Notice that whenever x is increasing, v is positive. In fact, this is almost tautologically necessary!

To elaborate on this point, see how x is increasing approximately on the intervals $$[-2.5,-0.5]$$ and $$[0.5,2.5]$$. These are the intervals where the red curve is going up.

Correspondingly, v is positive on these same intervals. This is where the blue curve is above the t-axis.

Of course where x is decreasing, v is negative.

When x switches from increasing to decreasing, we call this a "local maximum". This graph shows two different local maxima, at $$t\approx -0.5$$, and at $$t\approx 2.5$$.

When x switches from decreasing to increasing, we call this a "local minimum". This graph shows two local minima at $$t\approx-2.5$$ and $$t\approx-0.5$$.

Graph the function $$x(t) = t^2$$ and identify the intervals on which it is increasing and decreasing.

Suppose that x is increasing on the interval a to b. (a or b may be positive or negative infinity.)

Use this to draw a very rough sketch of what v could look like, by making v positive on the interval from a to b and negative everywhere else.

v is, of course, the velocity function, and x is the position function. Both model the motion of the same object, but by measuring different quantities.

(The graph that you draw does not need to be very precise at all, and many different graph shapes are plausible. We will later on learn to be more precise and quantitative about the graph of v which results from x.  For now it is necessary that we get the sign correct, but nothing else needs to be correct.)

Suppose that an object has velocity function $$v(t) = t^2$$. Use this to give a rough sketch of $$x(t)$$. As usual, x is position and v is velocity, for the same object.

Do so by merely identifying the intervals at which v is positive, negative, and zero. Wherever v is positive, draw x increasing. Wherever v is negative, draw x decreasing. Wherever v is zero, have x "leveling off" in some sense that you should try to reason through on your own.

In the diagram to the right, two functions are plotted. (They are the graphs of sine and cosine, if you must know! But that really should not be important for this exercise.)

One of these graphs is a position function and the other is the velocity function for some object. Which is which?

Let us make official some of the language that we have been using.

Show that $$t=0$$ is a local minimum preimage of $$x(t) = t^2$$.

Find the local maximum point for x.

Find all local maxima and minima in the graph to the right.

Suppose that you throw a rock straight up into the air.

1. At the moment when you release the rock, its height is increasing and therefore its velocity is.

2. After it travels upward for a time, it starts to slow. Then it turns around, and starts falling back toward the ground. Its velocity is now negative, which is the same as the fact that its position is.

3. Because the rock was moving upward, then at some point started moving downward, therefore the velocity must have switched from positive to negative at some point. When it was precisely at that point where the switch occurred, the velocity must have been exactly.

4. At the moment when the rock reached its maximum height, the velocity of the rock was.

Acceleration Functions
Because we are interested in velocity functions which change over time, we also want to apply our △ notation to velocity as well. The rate of change of velocity is called the "acceleration" of an object.

Because we often use the symbol a for acceleration, then we should no longer use it in a time interval like [a,b], as we did before.

Therefore we reserve the symbol a for acceleration, and instead talk about time intervals like $$[\alpha,\beta]$$.

Note that the definition above is a near perfect mirror of the definition of velocity, when related to position and time.

We should also give a moment of consideration to the units of acceleration. The units of $$\Delta_\alpha^\beta v$$ are the units of velocity, which are m/s. The unit of $$\Delta_\alpha^\beta t$$ is still the unit of time, s.

Therefore the units of a are $$\frac{m/s}{s}$$. This is more simply written as $$\frac{m}{s^2}$$ and accordingly pronounced "meters per second squared".

Suppose an object has velocity function, v, which decreases at a constant rate of $$-9.8\ \ m/s^2$$. Suppose that its initial velocity was 60 m/s, after which 5 seconds elapse. What is its final velocity?

Constant Acceleration


In the experiment above, we concluded that objects do not free fall at a constant speed. But if the speed is not constant, then what is it?

It seems clear that the speed increases, but increases in what way, exactly?

The simplest guess is to assume that the speed increases "constantly". That is to say, we might guess that the object's acceleration is constant.

Constant acceleration is animated in the image to the right. The animation shows, at the far left, the motion of an object falling straight down. Beside that is a graph of its position function, which is decreasing the whole time, but decreases faster and faster.

To the right of that, is a graph of the velocity function. It starts at 0 (i.e. $$v(0)=0$$) and thereafter is always negative. Moreover, it decreases at a constant rate.

To the right of velocity, finally, is the constant acceleration graph. It is perfectly horizontal, representing the fact that the acceleration does not change.

Note that constant acceleration implies a linear velocity graph. But the challenge for us is to go from the linear velocity graph, to the curved position graph.

Is this a reasonable guess as to how free fall motion really works? Again, let's make this official by declaring a hypothesis.

If an object is released into free fall, it will fall with a constant acceleration.

It is now significantly harder to test this hypothesis. If we collect a bunch of measurements of position and time for the free fall of an object, it will be very hard to know whether they confirm the hypothesis or not.

We would be much more capable of testing the constant acceleration hypothesis, if we could figure out the position function from the velocity function. Therefore this will be the subject of our study in the next lesson.