“Ladies and gentlemen, may I direct your attention to the center ring. Witness before you two ordinary textbooks, one labeled College Physics and the other Calculus. Their combined 2,500+ pages weigh over 25 lbs. Yet in this chapter and the next, your brave stunt-authors will attempt a most death-defying and impossible spectacle of mysticism and subterfuge: to reduce these two massive books into a mere 150 pages!”
Just like any good circus act, this one is prefaced with a lot of build up to set your expectations. The difference here is that the purpose of our preface is to lower your expectations.
OK, there's no way we can really cover all of physics and calculus in two chapters. As any politician knows, the secret to effectively communicate complicated subject matter in a short amount of time is to use lies, both the omission and commission kind. Let's talk about each of these kinds of lies in turn, so you will know what's really in store.
Just about everything—let's talk about what we are leaving out of physics first. To put the word “physics” on this chapter would be even more of an insult to people who do real physics than this chapter already is. We are concerned only with mechanics, and very simple mechanics of rigid bodies at that. Some topics traditionally found in a first-year physics textbook that are not discussed in this book include:
A note about energy and work is in order, because even in the limited context of mechanics, the fundamental concept of energy plays a central role in traditional presentations. Many problems are easier to solve by using conservation of energy than by considering the forces and applying Newton's laws. (In fact, an alternative to the Newtonian dynamics that we study in this book exists. It is known as Lagrangian dynamics and focuses on energy rather than forces. When used properly, both systems produce the same results, but Lagrangian dynamics can solve certain problems more elegantly and is especially adept at handling friction, compared to Newtonian dynamics.) However, at the time of this writing, basic general purpose digital simulations are based on Newtonian dynamics, and energy does not play a direct role. That isn't to say an understanding of energy is useless; indeed disobedience of the conservation of energy law is at the heart of many simulation problems! Thus, energy often arises more as a way to understand the (mis)behavior of a digital simulation, even if it doesn't appear in the simulation code directly.
Now let's talk about the ways in which this book will irritate calculus professors. We think that a basic understanding of calculus is really important to fully grasp many of the concepts from physics. Conversely, physics provides some of the best examples for explaining calculus. Calculus and physics are often taught separately, usually with calculus coming first. It is our opinion that this makes calculus harder to learn, since it robs the student of the most intuitive examples—the physics problems for which calculus was invented to solve! We hope interleaving calculus with physics will make it easier for you to learn calculus.
Our calculus needs are extremely modest in this book, and we have left out even more from calculus than we did from physics. After reading this chapter, you should know:
Of course, we are aware that a large number of readers may already have this knowledge. Take a moment to put yourself into one of the following categories:
Level 2 knowledge of calculus is sufficient for this book, and our goal is to move everybody who is currently in category 1 into category 2. If you're in category 3, our calculus discussions will be a (hopefully entertaining) review.We have no delusions that we can move anyone who is not already there into category 3.
The universe is commonly thought to be discrete in both space and time. Not only is matter broken up into discrete chunks called atoms, but there is evidence that the very fabric of space and time is broken up into discrete pieces also. Now, there is a difference of opinion as to whether it's really that way or just appears that way because the only way we can interact with space is to throw particles at it, but it's our opinion that if it looks like a duck, walks like a duck, quacks like a duck, has webbed feet and a beak, then it's a good working hypothesis that it tastes good when put into eggrolls with a little dark sauce.
For a long time, the mere thought that the universe might not be continuous had not even considered the slightest possibility of crossing anybody's mind, until the ancient Greeks got a harebrained and totally unjustified idea that things might be made up of atoms. The fact that this later turned out to be true is regarded by many as being good luck rather then good judgment. Honestly, who would have thought it? After all, everyday objects, such as the desk on which one of the authors is currently resting his wrists as he types this sentence, give every appearance of having smooth, continuous surfaces. But who cares? Thinking of the desk as having a smooth, continuous surface is a harmless but useful delusion that lets the author rest his wrists comfortably without worrying about atomic bond energy and quantum uncertainty theory at all.
Not only is this trick of thinking of the world as continuous a handy psychological rationalization, it's also good mathematics. It turns out that the math of continuous things is a lot less unwieldy than the math of discrete things. That's why the people who were thinking about how the world works in the 15th century were happy to invent a mathematics for a continuous universe; experimentally, it was a good approximation to reality, and theoretically the math worked out nicely. Sir Isaac Newton was thus able to discover a lot of fundamental results about continuous mathematics, which we call “calculus,” and its application to the exploration of a continuous universe, which we call “physics.”
Now, we're mostly doing this so that we can model a game world inside a computer, which is inherently discrete. There's a certain amount of cognitive dissonance involved with programming a discrete simulation of a continuous model of a discrete universe, but we'll try not to let it bother us. Suffice it to say that we are in complete control of the discrete universe inside our game, and that means that we can choose the kind of physics that applies inside that universe. All we really need is for the physical laws to be sufficiently like the ones we're used to for the player to experience willing suspension of disbelief, and hopefully say, “Wow! Cool!” and want to spend more money. For almost all games that means a cozy Newtonian universe without the nasty details of quantum mechanics or relativity. Unfortunately, that means also that there are a pair of nasty trolls lurking under the bridge, going by the names of chaos and instability, but we will do our best to appease them.
For the moment, we are concerned about the motion of a small object called a “particle.” At any given moment, we know its position and velocity.1 The particle has mass. We do not concern ourselves with the orientation of the particle (for now), and thus we don't think of the particle as spinning. The particle does not have any size, either. We will defer adding those elements until later, when we shift from particles to rigid bodies.
We are studying classical mechanics, also known as Newtonian mechanics, which has several simplifying assumptions that are incorrect in general but true in everyday life in most ways that really matter to us. So we can darn well make sure they are true inside our computer world, if we please. These assumptions are:
The first two are shattered by relatively, and the second two by quantum mechanics. Thankfully, these two subjects are not necessary for video games, because your authors do not have more than a pedestrian understanding of them.
We will begin our foray into the field of mechanics by learning about kinematics, which is
the study of the equations that describe the motion of a particle in various simple but
commonplace situations. When studying kinematics, we are not concerned with the causes of
motion—that is the subject of
dynamics, which will be covered in Chapter 12. For now, “ours is not to
question why,” ours is just to do the math to get equations that predict the position, velocity,
and acceleration of the particle at any given time
Because we are treating our objects as particles and tracking their position only, we will not consider their orientation or rotational effects until Chapter 12. When rotation is ignored, all of the ideas of linear kinematics extend into 3D in a straightforward way, and so for now we will be limiting ourselves to 2D (and 1D). This is convenient, since the authors do not know how to design those little origami-like things that lay flat and then pop up when you open the book, and the publisher wouldn't let us even if we were compulsive enough to learn how to do it. Later we'll see why treating objects as particles is perfectly justifiable.
Mechanics is concerned with the relationship among three fundamental quantities in nature: length, time, and mass. Length is a quantity you are no doubt familiar with; we measure length using units such as centimeters, inches, meters, feet, kilometers, miles, and astronomical units.2 Time is another quantity we are very comfortable with measuring, in fact most of us probably learned how to read a clock before we learned how to measure distances.3 The units used to measure time are the familiar second, minute, day, week, fortnight,4 and so on. The month and the year are often not good units to use for time because different months and years have different durations.
The quantity mass is not quite as intuitive as length and time. The measurement of an object's mass is often thought of as measuring the “amount of stuff” in the object. This is not a bad (or at least, not completely terrible) definition, but its not quite right, either . A more precise definition might be that mass is a measurement of inertia, that is, how much resistance an object has to being accelerated. The more massive an object is, the more force is required to start it in motion, stop its motion, or change its motion.
Mass is often confused with weight, especially since the units used to measure mass are also used to measure weight: the gram, pound, kilogram, ton, and so forth. The mass of an object is an intrinsic property of an object, whereas the weight is a local phenomenon that depends on the strength of the gravitational pull exerted by a nearby massive object. Your mass will be the same whether you are in Chicago, on the moon, near Jupiter, or light-years away from the nearest heavenly body, but in each case your weight will be very different. In this book and in most video games, our concerns are confined to a relatively small patch on a flat Earth, and we approximate gravity by a constant downward pull. It won't be too harmful to confuse mass and weight because gravity for us will be a constant. (But we couldn't resist a few cool exercises about the International Space Station.)
In many situations, we can discuss the relationship between the fundamental quantities without
concern for the units of measurement we are using. In such situations, we'll find it useful to
denote length, time, and mass by
For example, we might express a measurement of area as a number of “square feet.” We have
created a unit that is in terms of another unit. In physics, we say that a measurement of area
has the unit “length squared,” or
One last example is frequency. You probably know that frequency measures how many times
something happens in a given time interval (how “frequently” it happens). For example, a
healthy adult has an average heart rate of around 70 beats per minute (BPM). The motor in a car
might be rotating at a rate of 5,000 revolutions per minute (RPM). The
NTSC television standard is defined as 29.97 frames per second (FPS). Note that in each of
these, we are counting how many times something happens within a given duration of time. So we
can write frequency in generic units as
Table 11.1 summarizes several quantities that are measured in physics, their relation to the fundamental quantities, and some common units used to measure them.
|Quantity||Notation||SI unit||Other units|
Energy / Time
Of course, any real measurement doesn't make sense without attaching specific units to it. One
way to make sure that your calculations always make sense is to carry around the units at all
times and treat them like algebraic variables. For example, if you are computing a pressure and
your answer comes out with the units m/s, you know you have done something wrong; pressure has
units of force per unit area, or
Because unit conversion is an important skill, let's briefly review it here. The basic concept is
that to convert a measurement from one set of units to another, we multiply that measurement by a
well-chosen fraction that has a value of 1. Let's take a simple example: how many feet is
14.57 meters? Looking up the conversion factor,5 we see that
Our conversion factor tells us that the numerator and denominator of the fraction in Equation (11.1) are equal: 3.28083 feet is equal to 1 meter. Because the numerator and denominator are equal, the “value” of this fraction is 1. (In a physical sense, though, certainly numerically the fraction doesn't equal 1.) And we know that multiplying anything by 1 does not change its value. Because we are treating the units as algebraic variables, the m on the left cancels with the m in the bottom of the fraction.
Of course, applying one simple conversion factor isn't too difficult, but consider a more complicated example. Let's convert 188 km/hr to ft/s. This time we need to multiply by “1” several times:
We begin our study of kinematics by taking a closer look at the simple concept of speed. How do we measure speed? The most common method is to measure how much time it takes to travel a fixed distance. For example, in a race, we say that the fastest runner is the one who finishes the race in the shortest amount of time.
Consider the fable of the tortoise and the hare. In the story, they decide to have a race, and the hare, after jumping to an early lead, becomes overconfident and distracted. He stops during the race to take a nap, smell the flowers, or some other form of lollygagging. Meanwhile, the tortoise plods along, eventually passing the hare and crossing the finish line first. Now this is a math book and not a self-help book, so please ignore the moral lessons about focus and perseverance that the story contains, and instead consider what it has to teach us about average velocity. Examine Figure 11.1, which shows a plot of the position of each animal over time.
A play-by-play of the race is as follows. The gun goes off at time
To measure the average velocity of either animal during any time interval, we divide the
animal's displacement by the duration of the interval. We'll be focusing on the hare, and we'll
denote the position of the hare as
This is the definition of average velocity. No matter what specific units we use, velocity
always describes the ratio of a length divided by a time, or to use the notation discussed in
Section 11.2, velocity is a quantity with units
If we draw a straight line through any two points on the graph of the hare's position, then the
slope of that line measures the average velocity of the hare over the interval between the two
points. For example, consider the average velocity of the hare as he decelerates from time
Returning to Figure 11.1, notice that the hare's average velocity from
Average velocity can also be zero, as illustrated during the hare's nap from
And, of course, the final lesson of the fable is that the average velocity of the tortoise is
greater than the average velocity of the hare, at least from
One last thing to point out. If we assume the hare learned his lesson and congratulated the
tortoise (after all, let's not attribute to the poor animal all the negative personality
traits!), then at
We've seen how physics defines and measures the average velocity of an object over an interval,
that is, between two time values that differ by some finite amount
don't work when we are considering only a single instant in time. What are
Because of the vast array of problems to which the derivative can be applied, Newton was not the only one to investigate it. Primitive applications of integral calculus to compute volumes and such date back to ancient Egypt. As early as the 5th century, the Greeks were exploring the building blocks of calculus such as infinitesimals and the method of exhaustion. Newton usually shares credit with the German mathematician Gottfried Leibniz6 (1646–1716) for inventing calculus in the 17th century, although Persian and Indian writings contain examples of calculus concepts being used. Many other thinkers made significant contributions, includingFermat, Pascal, and Descartes.7 It's somewhat interesting that many of the earlier applications of calculus were integrals, even though most calculus courses cover the “easier” derivative before the “harder” integral.
We first follow in the steps of Newton and start with the physical example of velocity, which we feel is the best example for obtaining intuition about how the derivative works. Afterwards, we consider several other examples where the derivative can be used, moving from the physical to the more abstract.
Back to the question at hand: how do we measure instantaneous velocity? First, let's observe one particular situation for which it's easy: if an object moves with constant velocity over an interval, then the velocity is the same at every instant in the interval. That's the very definition of constant velocity. In this case, the average velocity over the interval must be the same as the instantaneous velocity for any point within that interval. In a graph such as Figure 11.1, it's easy to tell when the object is moving at constant velocity because the graph is a straight line. In fact, almost all of Figure 11.1 is made up of straight line segments,8 so determining instantaneous velocity is as easy as picking any two points on a straight-line interval (the endpoints of the interval seem like a good choice, but any two points will do) and determining the average velocity between those endpoints.
But consider the interval from
For concreteness in this example, let's assign some particular numbers. To keep those numbers
round (and also to stick with the racing theme), please allow the whimsical choice to measure
time in minutes and distance in
furlongs.9 We will assign
It's not immediately apparent how we might measure or calculate the velocity at the exact
In Figure 11.5, we fix the left endpoint of a line segment at
Let's carry out this experiment with some real numbers and see if we cannot approximate the instantaneous velocity of the hare. In order to do this, we'll need to be able to know the position of the hare at any given time, so now would be a good time to tell you that the position of the hare is given by the function11
Table 11.2 shows tabulated calculations for average
velocity over intervals with a right hand endpoint
The right-most column, which is the average velocity, appears to be converging to a velocity of
1 furlong/minute. But how certain are we that this is the correct value? Although we do not
have any calculation that will produce a resulting velocity of exactly 1 furlong/minute, for all
practical purposes, we may achieve any degree of accuracy desired by using this approximation
technique and choosing
This is a powerful argument. We have essentially assigned a value to an expression that we
cannot evaluate directly. Although it is mathematically illegal to substitute
Convergence arguments such as this are defined with rigor in calculus by using a formalized tool known as a limit. The mathematical notation for this is
The notation `
In general, an expression of the form
This is an important idea, as it defines what we mean by instantaneous velocity.
We won't have much need to explore the full power of limits or get bogged down in the finer points; that is the mathematical field of analysis, and would take us a bit astray from our current, rather limited,12 objectives. We are glossing over some important details13 so that we can focus on one particular case, and that is the use of limits to define the derivative.
The derivative measures the rate of change of a function. Remember that
“function” is just a fancy word for any formula, computation, or procedure that takes an input
and produces an output. The derivative quantifies the rate at which the output of the function
will change in response to a change to the input. If
For now, we are in an imaginary racetrack where rabbits and turtles race and moral lessons are
taught through metaphor. We have a function with an input of
When we calculate a derivative, we won't end up with a single number. Expecting the answer to “What is the velocity of the hare?” to be a single number makes sense only if the velocity is the same everywhere. In such a trivial case we don't need derivatives, we can just use average velocity. The interesting situation occurs when the velocity varies over time. When we calculate the derivative of a position function in such cases, we get a velocity function, which allows us to calculate the instantaneous velocity at any point in time.
The previous three paragraphs express the most important concepts in this section, so please allow us to repeat them.
The next few sections discuss the mathematics of derivatives in a bit more detail, and we return to kinematics in Section 11.5. This material is aimed at those who have not had14 first-year calculus. If you already have a calculus background, you can safely skip ahead to Section 11.5 unless you feel in need of a refresher.
Section 11.4.2 lists several examples of derivatives to give you a better
understanding of what it means to measure a rate of change, and also to back up our claim that
the derivative has very broad applicability. Section 11.4.3
gives the formal mathematical definition of the derivative15 and shows how to use this definition to solve problems. We also
finally figure out how fast that hare was moving at
Velocity may be the easiest introduction to the derivative, but it is by no means the only example. Let's look at some more examples to give you an idea of the wide array of problems to which the derivative is applied.
The simplest types of examples are to consider other quantities that vary with time. For
There are also physical examples for which the independent variable is not time. The prototypical
case is a function
Now let's become a bit more abstract, but still keep a physical dimension as the independent
variable. Let's say that for a popular rock-climbing wall, we know a function
Now consider the interpretation of derivative
One last example. Figure 11.6 shows happiness as a function of salary. In this case, the derivative is essentially the same thing as what economists would call “marginal utility.” It's the ratio of additional units of happiness per additional unit of income. According this figure, the marginal utility of income decreases, which of course is the famous law of diminishing returns. According to our research,19 it even becomes negative after a certain point, where the troubles associated with high income begin to outweigh the psychological benefits. The economist-speak phrase “negative marginal utility” is translated into everyday language as “stop doing that.”
Now we're ready for the official20 definition of the derivative found in most math textbooks, and
to see how we can compute derivatives using the definition. A derivative can be understood as
the limiting value of
Here the notation for the derivative
In certain circumstances, infinitesimals may be manipulated like algebraic variables (and you can
also attach units of measurement to them and carry out dimensional analysis to check your work).
The fact that such manipulations are often correct is what gives Leibniz notation its intuitive
appeal. However, because they are infinitely small values, they require special handling, similar
to the symbol
Differentiating a simple function by using the definition Equation (11.3) is an important rite of passage, and we are proud to help you cross this threshold. The typical procedure is this:
Applying this procedure to our case yields
Now we are at step 3. Taking the limit in
Equation (11.4) is now easy; we simply substitute
Finally! Equation (11.5) is the velocity
function we've been looking for. It allows us to plug in any value of
So the instantaneous velocity of the hare at
Figure 11.7 shows this point and several others along the interval we've been studying. For each point, we have calculated the instantaneous velocity at that point according to Equation (11.5) and have drawn the tangent line with the same slope.
It's very instructive to compare the graphs of position and velocity side by side. Figure 11.8 compares the position and velocity of our fabled racers.
There are several interesting observations to be made about Figure 11.8.
At this point, we should acknowledge a few ways in which our explanation of the derivative differs from most calculus textbooks. Our approach has been to focus on one specific example, that of instantaneous velocity. This has led to some cosmetic differences, such as notation. But there were also many finer points that we are glossing over. For example, we have not bothered defining continuous functions, or given rigorous definitions for when the derivative is defined and when it is not defined. We have discussed the idea behind what a limit is, but have not provided a formal definition or considered limits when approached from the left and right, and the criteria for the existence of a well-defined limit. We feel that leading off with the best intuitive example is always the optimum way to teach something, even if it means “lying” to the reader for a short while. If we were writing a calculus textbook, at this point we would back up and correct some of our lies, reviewing the finer points and giving more precise definitions.
However, since this is not a calculus textbook, we will only warn you that what we said above is the big picture, but isn't sufficient to handle many edge cases when functions do weird things like go off into infinity or exhibit “jumps” or “gaps.” Fortunately, such edge cases just don't happen too often for functions that model physical phenomena, and so these details won't become an issue for us in the context of physics.
We do have room, however, to mention alternate notations for the derivative that you are likely to encounter.
Several different notations for derivatives are in common use. Let's point out some ways that
other texts might look different from what we've said here. First of all, there is a trivial
issue of naming. Most calculus textbooks define the derivative in very general terms, where the
output variable is named
A variation on the Leibniz notation we prefer in this book is to prefix an
can be read as “the derivative with respect to
It's important to interpret these manipulations as notational manipulations rather than having any real mathematical meaning. The notation is attractive because such algebraic manipulations with the infinitesimals often work out. But we reiterate our warning to avoid attaching much mathematical meaning to such operations.
Another common notation is to refer to the derivative of a function
notation. It's used when the independent variable that
we are differentiating with respect to is implied or understood by context. Using this notation
we would define velocity as the derivative of the position function by
One last notation, which was invented by Newton and is used mostly when the independent variable
is time (such as in the physics equations Newton invented), is dot notation. A derivative
is indicated by putting a dot over the variable; for example,
Here is a summary of the different notations for the derivative you will see, using velocity and position as the example:
Now let's return to calculating derivatives. In practice, it's seldom necessary to go back to
the definition of the derivative in order to differentiate an expression. Instead, there are
simplifying rules that allow you to break down complicated functions into smaller pieces that can
then be differentiated. There are also special functions, such as
In this book, our concerns are limited to the derivatives of a very small set of functions, which luckily can be differentiated with just a few simple rules. Unfortunately, we don't have the space here to develop the mathematical derivations behind these rules, so we are simply going to accompany each rule with a brief explanation as to how it is used, and a (mathematically nonrigorous) intuitive argument to help you convince yourself that it works.
Our first rule, known as the constant rule, states that the derivative of a constant
function is zero. A constant function is a function that always produces the same value. For
The next rule, sometimes known as the sum rule, says that differentiation is a linear operator. The meaning of “linear” is essentially identical to our definition given in Chapter 5, but let's review it in the context of the derivative. To say that the derivative is a linear operator means two things. First, to take the derivative of a sum, we can just take the derivative of each piece individually, and add the results together. This is intuitive—the rate of change of a sum is the total rate of change of all the parts added together. For example, consider a man who moves about on a train. His position in world space can be described as the sum of the train's position, plus the man's position in the body space of the train.23 Likewise, his velocity relative to the ground is the sum of the train's velocity relative to the ground, plus his velocity relative to the train.
The second property of linearity is that if we multiply a function by some constant, the derivative of that function gets scaled by that same constant. One easy way to see that this must be true is to consider unit conversions. Let's return to our favorite function that yields a hare's displacement as a function of time, measured in furlongs. Taking the derivative of this function with respect to time yields a velocity, in furlongs per minute. If somebody comes along who doesn't like furlongs, we can switch from furlongs to meters, by scaling the original position function by a factor of 201.168. This must scale the derivative by the same factor, or else the hare would suddenly change speed just because we switched to another unit.
The linear property of the derivative is very important since it allows us to break down many common functions into smaller, easier pieces.
One of the most important and common functions that needs to be differentiated also happens to be the easiest: the polynomial. Using the linear property of the derivative, we can break down, for example, a fourth-degree polynomial with ease:
The last derivative
This rule gives us the answers to the four derivatives needed above:
Notice in the last equation we used the identity
One last comment before we plug these results into
Equation (11.9) to differentiate our
polynomial. Using the identity
Let's get back to our fourth-degree polynomial. With the sum and power rule at our disposal, we can make quick work of it:
Below are several more examples of how the power rule can be used. Notice that the power rule works for negative exponents as well:
This section looks at some very special examples of differentiating polynomials. Given any
This pattern continues forever; in other words, to compute the exact value of
This is exactly the process by which trigonometric functions are computed inside a computer.
First, trig identities are used to get the argument into a restricted range (since the functions
are periodic). This is done because when the Taylor series is truncated, its accuracy is highest
near a particular value of
All this trivia concerning approximations is interesting, but our real reason for bringing up
Taylor series is to use them as nontrivial examples of differentiating polynomials with the power
rule, and also to learn some interesting facts about the sine, cosine, and exponential functions.
Let's use the power rule to differentiate the Taylor series expansion of
In the above derivation, we first used the sum rule, which says that to differentiate the whole
Taylor polynomial, we can differentiate each term individually. Then we applied the power rule
to each term, in each case multiplying by the exponent and decrementing it by one. (And also
Does Equation (11.11) the last look familiar? It should, because it's
the same as Equation (11.10), the Taylor series for
The derivatives of the sine and cosine functions will become useful in later sections.
Now let's look at one more important special function that will play an important role later in
this book, which will be convenient to be able to differentiate, and which also happens to have a
nice, tidy Taylor series. The function we're referring to is the exponential function,
Taking the derivative gives us
But this result is equivalent to the definition of
It is this special property about the exponential function that makes it unique and causes it to come up so frequently in applications. Anytime the rate of change of some value is proportionate to the value itself, the exponential function will almost certainly arise somewhere in the math that describes the dynamics of the system.
The example most of us are familiar with is compound interest. Let
You might have noticed that the Taylor series of
We hope this brief encounter with Taylor series, although a bit outside of our main thrust, has sparked your interest in a mathematical tool that is highly practical, in particular for its fundamental importance to all sorts of approximation and numerical calculations in a computer. We also hope it was an interesting non-trivial example of differentiation of a polynomial. It also has given us a chance to discuss the derivatives of the sine, cosine, and exponential functions; these derivatives come up again in latersections.
The chain rule is the last rule of differentiation we discuss here. The chain rule tells us how to determine the rate of change of a function when the argument to that function is itself some other function we know how to differentiate.
In the race between the tortoise and hare, we never really thought much about exactly what our
Let's say that we have a function
Now consider the composite function
This rule is known as the chain rule. It is particularly intuitive when written in
Leibniz notation, because the
Here are a few examples, using functions we now know how to differentiate:Examples of the
We're going to put calculus from a purely mathematical perspective on the shelf for a while and return our focus to kinematics. (After all, our purpose in discussing calculus was, like Ike Newton, to improve our understanding of mechanics.) However, it won't be long before we will return to calculus with the discussion of the integral and the fundamental theorem of calculus.
We've made quite a fuss about the distinction between instantaneous velocity and average
velocity, and this distinction is important (and the fuss is justified) when the velocity is
changing continuously. In such situations, we might be interested to know the rate at
which the velocity is changing. Luckily we have just learned about the derivative, whose
raison d'être is to investigate rates of change.
When we take the derivative of a velocity function
In ordinary conversation, the verb “accelerate” typically means “speed up.” However, in physics, the word “acceleration” carries a more general meaning and may refer to any change in velocity, not just an increase in speed. In fact, a body can undergo an acceleration even when its speed is constant! How can this be? Velocity is a vector value, which means it has both magnitude and direction. If the direction of the velocity changes, but the magnitude (its speed) remains the same, we say that the body is experiencing an acceleration. Such terminology is not mere nitpicking with words, the acceleration in this case is a very real sensation that would be felt by, say, two people riding in the back seat of a swerving car who find themselves pressed together to one side. We have more to say about this particular situation in Section 11.8.
We can learn a lot about acceleration just by asking ourselves what sort of units we should use
to measure it. For velocity, we used the generic units of
For example, an object in free fall near Earth's surface accelerates at a rate of about
More generally, the velocity at an arbitrary time
You should study Figure 11.9 until it makes sense to you. In particular, here are some noteworthy observations:
The accelerations experienced by an object can vary as a function of time, and indeed we can continue this process of differentiation, resulting in yet another function of time, which some people call the “jerk” function. We stick with the position function and its first two derivatives in this book. Furthermore, it's very instructive to consider situations in which the acceleration is constant (or at least has constant magnitude). This is precisely what we're going to do in the next few sections.
Section 11.6 considers objects under constant acceleration, such as objects in free fall and projectiles. This will provide an excellent backdrop to introduce the integral, the complement to the derivative, in Section 11.7. Then Section 11.8 examines objects traveling in a circular path, which experience an acceleration that has a constant magnitude but a direction that changes continually and always points towards the center of the circle.
Let's look now at the trajectory an object takes when it accelerates at a constant rate over time. This is a simple case, but a common one, and an important one to fully understand. In fact, the equations of motion we present in this section are some of the most important mechanics equations to know by heart, especially for video game programming.
Before we begin, let's consider an even simpler type of motion—motion with constant velocity. Motion with constant velocity is a special case of motion with constant acceleration—the case where the acceleration is constantly zero. The motion of a particle with constant velocity is an intuitive linear equation, essentially the same as Equation (9.1), the equation of a ray. In one dimension, the position of a particle as a function of time is
Now let's consider objects moving with constant acceleration. We've already mentioned at least
one important example: when they are in free fall, accelerating due to gravity. (We'll ignore
wind resistance and all other forces.) Motion in free fall is often called
projectile motion. We start out in one dimension here to keep things simple. Our goal is a
Take our example of illegal ball-bearing-bombing off of Willis Tower. Let's set a reference
At this point, we don't even know what form
Let's make our example a bit more specific. Earlier, we computed that after being in free fall
for 2.4 seconds, the ball bearing would have a velocity of
Table 11.3 shows tabulated values for 6, 12, and 24
slices. For each slice,
Since each slice has a different initial velocity, we are accounting for the fact that the velocity changes over the entire interval. (In fact, the computation of the starting velocity for the slice is not an approximation—it is exact.) However, since we ignore the change in velocity within a slice, our answer is only an approximation. Taking more and more slices, we get better and better approximations, although it's difficult to tell to what value these approximations are converging. Let's look at the problem graphically to see if we can gain some insight.
In Figure 11.10, each rectangle represents one time interval in our approximation. Notice that the distance traveled during an interval is the same as the area of the corresponding rectangle:
Now we come to the key observation. As we increase the number of slices, the total area of the rectangles becomes closer and closer to the area of the triangle under the velocity curve. In the limit, if we take an infinite number of rectangles, the two areas will be equal. Now, since total displacement of the falling ball bearing is equal to the total area of the rectangles, which is equal to the area under the curve, we are led to an important discovery.
We have come to this conclusion by using a limit argument very similar to the one we made to define instantaneous velocity—we consider how a series of approximations converges in the limit as the approximation error goes to zero.
Notice that we have made no assumptions in this argument about
procedure will work for any arbitrary velocity
function.28 This limit argument is a formalized tool in
calculus known as the
Riemann integral, which we will consider in Section 11.7. That will also be the
appropriate time to consider the general case of any
Remember the question we're trying to answer: how far does an object travel after being dropped
at an initial zero velocity and then accelerated due to gravity for 2.4 seconds at a constant
Thus after a mere 2.4 seconds, the ball bearing had already dropped more than 92 feet!
That solves the specific problem at hand, but let's be more general. Remember that the larger
goal was a kinematic equation
The rectangle has base
We have just derived a very useful equation, so let's highlight it so that people who are skimming will notice it.
It's common that we only need the displacement
Let's work through some examples to show the types of problems that can be solved by using
Equation (11.15) and its variants. One tempting scenario
is to let our ball bearing hit the ground. The observation deck on the 103rd floor of
Willis Tower is 1,353 ft above the sidewalk. If it is dropped from that height, how long will it
take to fall to the bottom?
Solving Equation (11.15) for
Equation (11.16) is a very useful general equation. Plugging in the numbers specific to this problem, we have
Naturally, a person in the business of dropping ball bearings from great heights is interested in how much damage he can do, so the next logical question is, “How fast is the ball bearing traveling when it hits the sidewalk?” To answer this question, we plug the total travel time into Equation (11.13):
If we ignore wind resistance, at the moment of impact, the ball bearing is traveling at a speed that covers a distance of roughly a football field in one second! You can see why the things we are doing in our imagination are illegal in real life. Let's keep doing them.
Now let's assume that instead of just dropping the ball bearing, we give it an initial velocity
(we toss it up or down). It was our free choice to decide whether up or down is positive in
these examples, and we have chosen
And now plugging in the numbers for our specific problem, we have
Notice that the result is negative, indicating an upwards velocity. If we give the ball bearing
this initial velocity, we might wonder how long it takes for the bearing to come back down to its
initial position. Using Equation (11.16) and
It's no surprise that
Examine the graph in Figure 11.12, which plots the position and
velocity of an object moving under constant velocity
The first observation is that the projectile reaches its maximum height, denoted
Right now we are in one dimension and considering only the height. But if we are in more than one dimension, only the velocity parallel to the acceleration must vanish. There could be horizontal velocity, for example. We discuss projectile motion in more than one dimension in just a moment.
The second observation is that the time it takes for the object to travel from its maximum
altitude to its initial altitude, denoted
The third and final observation is that the velocity at
Before we look at projectile motion in more than one dimension, let's summarize the formulas we have derived in this section. The first two are the only ones worth memorizing; the others can be derived from them.
Extending the ideas from the previous section into 2D or 3D is mostly just a matter of switching
to vector notation;
Note that we didn't make a vector version of Equation (11.17); we'll get to that in a moment.
This seemingly trivial change in notation is actually hiding two rather deep facts. First, in the
algebraic sense, the vector notation is really just shorthand for sets of parallel scalar
We were able to leap from 1D to 3D mostly just by bolding a few letters due to the independence
of the coordinates. However, there is a bit more to say about projectile motion in multiple
dimensions because there are situations where we need to consider the effects of all the
coordinates at the same time. One situation has already been alluded to by the lack of a vector
equation corresponding to Equation (11.17).
In other words, how could we solve for time
For example, earlier we computed how long it would take for a ball bearing dropped from a great height to hit the sidewalk below, which is a one-dimensional problem. The corresponding three-dimensional problem would be to try to drop the ball bearing into a bucket which is free to move around on the sidewalk. Let's say that the bucket is off to our left. Our initial velocity had better have some leftward component then, or else the ball bearing won't land in the bucket. Another indication that the multi-dimensional case is more complicated than 1D is that a direct translation of Equation (11.17) into vector form results in nonsensical operations of taking the square root of a vector and dividing one vector by another.
The key to solving this problem is to realize that any horizontal changes (either to the bucket's
position or the initial velocity of the ball bearing) do not affect how long it takes the
ball bearing to reach the sidewalk. This is because the coordinates are independent from one
another. The horizontal velocity and acceleration do not interact with the vertical velocity and
acceleration. To be specific, let's switch to our standard 3D coordinate system, which has
Let's talk a bit more about exactly what it means to “chose which direction to use,” as was stated in the previous paragraph. In cases of simple projectile motion, such as the ball-bearing example, where gravity is the constant acceleration, the direction to choose is obvious: use the direction of gravity. Furthermore, because coordinate systems are chosen such that “up” is one of the cardinal axes, the process of solving a one-dimensional problem in that direction is a trivial matter of plucking out the appropriate Cartesian coordinate and discarding the others. In general, however, the situation can be more complicated. But before we discuss the details of the general case, there are a few more things we can say about this very important and common special situation.
To study projectile motion where acceleration is solely due to gravity, which is a constant and
acts along a cardinal axis, let's establish a 2D coordinate space where
Reviewing the notation in Figure 11.13, we see that we can express the
position of the particle as a function of time either as
Let's state the equations for velocity and position using the notation just described:
The distances labeled
and at this time, the height is equal toAltitude at apex
We stated earlier that the time for the object to come back down to its initial height (which we
As expected, the flight time
We often wish to specify the initial velocity in terms of an angle and speed,
rather than velocities along each axes. In other words, we wish to use polar coordinates rather
than Cartesian. As shown in Figure 11.13, we denote the initial launch
We can also express
These equations are highly practical because they directly capture the relationship between the “user-friendly” quantities of launch speed, launch angle, flight time, and flight distance.
At this point, let's pause to make an interesting observation about the relationship between the
Now let's return to a question we put on hold from earlier: how might we determine the point of
impact for any arbitrary vectors
Assuming the projectile hits the target, we will get the same value for
We have just showed that the total displacement of an object in a time interval is equal to the area under the plot of the object's velocity. We used the example of constant acceleration, which has a simple graph, and the area was easy to solve geometrically. We did not pursue in further generality the limit argument that led us to the surprising equivalence, because this special case has such compelling applications. Now we are ready to discuss more general cases. The need to compute a “continuous summation,” where the rate of growth is a known function, is a common concept in engineering and science. The calculus tool used to compute these sums is the integral.
If you have already studied integral calculus and have a good intuition about what the integral is used for, then you can safely skip ahead to Section 11.8, when our focus returns to the subject of mechanics. However, if you've never had integral calculus or if your intuition about the integral is a bit shaky, keep reading.
There are two important ways of approaching the integral. The first way is essentially to make the notion of “summing up many tiny elements” a bit more precise and introduce some mathematical formalism. The other way is to compare the integral to the derivative. It's important to understand both interpretations. The integral is a bit more difficult to grasp than the derivative, but for reasons that become apparent later, it plays a much greater role in physics simulations and many other areas of video game programming. Understanding what the integral does is very important, even if the vast assortment of pen-and-paper techniques to compute integrals analytically is not very useful in our case, being replaced instead by techniques of numerical integration.
Let's turn our informal summation into mathematics notation, in which we compute the area under
The error in this approximation decreases as we increase the number of slices