Falling Stuff on Earth
Our last application is something I like to call "falling stuff on earth." This was one of the puzzles that got Newton and Liebniz thinking about this stuff during the Renaissance, so it has some nice historical significance. It also gives us an opportunity to look again at the fundamental concept of integration, review the FTC, and think about constants of integration.
We only need one fact from physics: when stuff falls on earth due to the force of gravity, it accelerates towards the center of the earth at about 9.8 meters-per-second per second.
I know the units of "meters-per-second per second" are odd, but just think of it this way: every second something free-falls on earth, the falling thing will fall 9.8 meters per second faster. For instance, if something starts at rest (in your hand for instance), after 1 second it will be falling -9.8 meters-per-second; after 2 seconds it will be falling at
-19.6=-9.8*2 meters-per-second; after 3 seconds it will be falling at -29.4=-9.8*3 meters-per-second; and so on. In short, multiply the number of seconds the thing has been falling with -9.8 meters-per-second per second, and you'll know the velocity of the falling thing in meters-per-second.
There's quite a few caveats to this rule about acceleration. For instance, when stuff falls on earth it doesn't endlessly accelerate. Indeed, when the falling stuff runs into something, it decelerates, sometimes rapidly. For example, when falling stuff hits the ground, it violently decelerates and comes to a stop. SPLAT! But falling stuff also runs into air, and that also affects the acceleration of the falling stuff, but more gently.
We're going to completely ignore all of these nuances here in Season 1 of Calculus for the People. We'll come back to them in Season 4. For now, I propose an extremely simplified model of the acceleration of falling stuff on earth:
a(x)=-9.8
In words, this model says the acceleration due to gravity is always minus 9.8 meters-per-second per second. In the language of atomic functions: acceleration of falling stuff on earth is modeled by a constant function.
The reason 9.8 is negative is because falling stuff on earth falls down. We could also do everything here with a positive 9.8 so long as we understand that down is positive, but traditionally, it's done with a negative 9.8. This choice also forces you to think about negative integrals which is good for you.
Now I ask you: can you use the model a(x) to construct a model of the velocity of the falling thing?
A first mistake people make is to think that because the acceleration is constant, that velocity is also a constant. But you already know that's not true. I said above "As time progresses, falling stuff will fall faster and faster under the influence of Earth's gravity."
So how do we model the velocity with a(x)? With an integral of course! Because a(x) is a mathematical model of a rate, its integral is a mathematical model of the accumulated effect of that rate: velocity. Confusingly a(x) is the rate of change of rate of change, but don't let this trip you up.
Check out the applet below illustrating this. On the left is
a(x)=-9.8, the function to be integrated. On the right is v(x)=-9.8x+v_0 the antiderivative of a(x). The units of v(x) are meters-per-second
Just as in previous activities, on the left, the red area is the integral of a(x), and on the right the red segment is the difference in two velocity readings. The Fundamental Theorem of Calculus says these are equal. At the outset, the applet indicates that both are -19.6 meters-per-second, the velocity of the falling thing after 2 seconds. In other words, this illustrates the calculation of velocity that we did earlier in this activity. Scroll up and read it again if you need to.
We're calling the antiderivative v(x) instead of A(x) because it is a model of velocity. Also the constant of integration is v_0 instead of c.
Speaking of constants of integration, let's think about what the meaning of the constant of integration is in this model. As I noted in the previous activity and as we can see in the applet above, the constant of integration doesn't impact the value of the integral of a(x). That said, the constant of integration does impact the meaning of the antiderivative, v(x).
In this model, the constant of integration v_0 represents the initial velocity of the object.
For instance, if the thing falling on earth is dropped from rest, then v_0 would be 0. If the object is tossed downwards and then gravity accelerates it, v_0 would be negative. If the object is tossed upwards and then gravity decelerates it, stops it momentarily, and then accelerates it downwards, v_0 would be positive.
Try it out! Slide v_0 above and notice it slides v(x) up and down. In particular, when v_0 is positive, v(x) at time 0 is positive indicating the object is tossed upwards (away from earth) at the start of the model. Similarly, when v_0 is negative, v(x) is negative at time 0 indicating the object is tossed downwards (towards earth) at the start of the model. But let's not stop here. Let's do another integral, and see what we can learn from integrating
v(x)=-9.8x+v_0. In the applet below v(x) is now the function to be integrated, and appears on the left, and its antiderivative s(x)=-9.8/2x^2+v_0x+s_0 is on the right. Note: -9.8/2 is -4.9.The antiderivative is called
s(x) because it represents the "position above earth" (or more concisely: height) in meters. I actually really hate this naming convention; I think this antiderivative should be called h(x) for height, but I defer to tradition here.
Either way, note that we can be sure s(x) is a model of height simply because height is the accumulated effect of velocity v(x). As the velocity becomes more and more negative in the left panel, the integral of that velocity represents the effect of that negative velocity, and that of course means the height of the object will get lower and lower as illustrated in the right panel.
What about the new constant of integration s_0? At the outset s_0 is set to 50. You now have sliders for both constants of integration. The slider for v_0 is on the left, and the slider for s_0 is on the right.
Question: What do you think s_0 being set to 50 represents?
Answer: The constant of integration s_0 in the antiderivative s(x) represents the initial height (in meters) of the falling thing. So with s_0 set to 50 and v_0 set to 0, s(x) is a model the height of a falling thing dropped gently (no initial velocity) from a height of 50 meters.
To help you build more intuition for this, I've put an almost identical copy of the "falling stuff" applet below, but with a new set of values for s_0 and v_0. You may recognize these numbers. They appeared in one of our very first activities when we modeled the height of an incoming missile.We've now come back to where we started, and I hope you appreciate how much more you know now than when you started!
The only difference from this and the model at the start of this book is that because we have used a model of acceleration due to gravity rounded to one decimal place,
a(x)=-9.8, the coefficient of x^2 has fewer decimal places here than it did at the start of the book.
But with your knowledge of integrals and derivatives, you now can see so much more depth in this model than you did when you started studying calculus. Congratulations!
An integral of v(x) from 0 to 20 tells you that the vertical displacement of the missile is 3996.95 meters UPWARDS in the first 20 seconds.
A Second Derivative Test of s(x) tells you that that time 30.39 seconds is a maximum height of the missile.
The code snippet intersect(s(x),y=0) will find two points, one of which is SPLAT!=(62.79,0) the time the model predicts the missile will land. You could do this by hand with the quadratic formula, but we of course won't do so in this course.
You also can now see s(x) as the result of a two-step integral analysis arising from just one fact from physics: that acceleration due to gravity on earth is constantly equal to negative 9.8 meters-per-second per-second.
Furthermore, you have a deep understanding of the numbers in this model now. 297.85 is the model's estimate of the initial vertical velocity, v_0, when it was first detected, and 616.06 is the initial height, s_0 , of the missile when it was first detected.
You should be proud of yourself! Getting here officially earns you the right to say that you understand Calculus.
In some sense, this is the logical end of this book. Conceptually, you now understand calculus. There's nothing more to do conceptually. That said, calculus does have some important calculations. For instance, the Monkey Rules for derivatives are an important calculation process that directly leads to conceptually understanding the integral.
Similarly, the process for finding antiderivatives will lead to conceptual understanding of topics in Season 2, Season 3 and Season 4 of Calculus for the People. To that end, we round out this chapter with a very brief overview of how to calculate antiderivatives.