Parametric Equations
Discussion
analytic geometry
In previous sections, we discussed how the kinematic variables of distance, displacement, speed, velocity, and acceleration can be used to analyze motion in a straight line. During one-dimensional motion, the objects moving have only one degree of freedom. This is an idealization, of course. When I step out of my apartment on to the sidewalk there is essentially one decision I have to make…
left or | right? |
But I'm not some sort of computer-driven robot forced by the definition of "sidewalk" to choose only between these two options. I also have the option to move forward or backward; that is, across the street or back into my apartment. This second degree of freedom makes walking a two-dimensional activity. If I walk far enough my altitude is also sure to change, so you might consider walking a three-dimensional activity, but this is merely a response to the Earth's surface.
Flying on the other hand is truly a three-dimensional activity; especially for helicopter pilots and hummingbirds. There are always three choices available to the people and animals that fly…
forward or | backward? |
left or | right? |
up or | down? |
or if you prefer…
north or | south? |
east or | west? |
up or | down? |
These choices are mutually exclusive of one another — a property that is known as orthogonality. Two directions are orthogonal if there is no way that motion along one of these directions could result in motion in any of the other directions. Walk east or west all you like, you'll never see any north-south change in your position. That's why walking is not normally considered a three-dimensional activity. The freedom to go "up" or "down" a hill is really just the freedom to choose between going "forward" or "backward". That one option results in upward motion while the other results in downward motion seems more a function of the surface of the Earth than in any choice on my part. I really can't decide to "turn up" while walking in the same way that I can decide to "turn left".
Orthogonal directions are always perpendicular to one another. Since "perpendicular" is a completely adequate word, "orthogonal" may seem unnecessarily pretentious. The thing with dimension is that it refers to more things than just the number of independent directions. Any two measurable things that are independent of one another can be considered dimensions. In thermodynamics (the study of heat and work) pressure and volume behave like up-down and left-right do in kinematics. The situation here is a bit more complicated than just saying "pressure is independent of volume", but it should be apparent that saying "pressure is perpendicular to volume" doesn't make any sense. It is correct to say that "pressure is orthogonal to volume", however.
Identifying the appropriate perpendicular directions in a kinematics problem is one of the first steps in solving it. Attach these directions (called axes) to a point in space (called the origin) and you've just created a coordinate system. There is no rule behind naming axes, but it is tradition to call the principle horizontal direction +x. Naming the other directions is open to debate, however. High school physics courses tend to concentrate on two-dimensional problems. An instructor may use +y for the perpendicular horizontal direction in one instance and then use +y for up in another. College professors tend to be more formal and usually reserve +y for the perpendicular horizontal direction and +z for up. I tend to think that if a situation is strictly two-dimensional then the other direction should be +y no matter what that direction may be. I only use +z when all three dimensions need to be considered. The thing with physics is that it doesn't really matter. The universe is isotropic. All the laws of physics are always true without modification no matter where you place the origin, how you orient the axes (as long as they're perpendicular, of course), or what you name them. Since it doesn't matter, why not make it easy on yourself? Place the origin wherever it's convenient and label the axes in whatever manner you wish. You don't even have to call them x, y, and z. Here are some alternate names for axes I have actually seen used…
a, b, and c
i, j, and k
horizontal and vertical
parallel (∥) and perpendicular (⊥)
r, θ, and φ
ζ and ξ
Since all perpendicular directions are orthogonal and since any vector quantity can be resolved into components along these directions, n-dimensional motion can be completely described by n one-dimensional algebraic expressions in n perpendicular directions (where n is any whole number greater than zero). Thus two-dimensional motion can be completely described by two one-dimensional algebraic expressions along two perpendicular directions — usually called x and y.
When a situation is essentially two-dimensional, the position of an object can be adequately described by two numbers (its x and y coordinates), the velocity of an object by two numbers (the x and y components of its velocity), and its acceleration by two numbers (the x and y components of its acceleration). From the Pythagorean theorem, the magnitudes of these quantities are related by the following expressions…
r^{2} = | x^{2} | + | y^{2} |
v^{2} = | v_{x}^{2} | + | v_{y}^{2} |
a^{2} = | a_{x}^{2} | + | a_{y}^{2} |
The relation is more completely described in vector notation where the vectors with a hat ( ̂) over them are unit vectors along the coordinate axes…
r = | x | î | + | y | ĵ |
v = | v_{x} | î | + | v_{y} | ĵ |
a = | a_{x} | î | + | a_{y} | ĵ |
Since kinematic vectors at right angles are independent of each other, associated with each direction are functions for displacement, average velocity, and average acceleration…
x = | x(t) | |
v̅_{x} = | Δx | |
Δt | ||
a̅_{x} = | Δv_{x} | |
Δt |
y = | y(t) | |
v̅_{y} = | Δy | |
Δt | ||
a̅_{y} = | Δv_{y} | |
Δt |
Similar functions can be derived for instantaneous velocity and instantaneous acceleration…
x = | x(t) | |
v_{x} = | lim | Δx |
Δt → 0 | Δt | |
a_{x} = | lim | Δv_{x} |
Δt → 0 | Δt |
y = | y(t) | |
v_{y} = | lim | Δy |
Δt → 0 | Δt | |
a_{y} = | lim | Δv_{y} |
Δt → 0 | Δt |
Or, in the language of calculus…
x = | x(t) | ||
v_{x} = | dx | ||
dt | |||
a_{x} = | dv_{x} | = | d^{2}x |
dt | dt^{2} |
y = | y(t) | ||
v_{y} = | dy | ||
dt | |||
a_{y} = | dv_{y} | = | d^{2}y |
dt | dt^{2} |
Likewise, three-dimensional motion can be completely described by three one-dimensional algebraic expressions along three mutually perpendicular directions usually called x, y, and z. Since the universe has three spatial dimensions, the position of an object can be completely described by three numbers (its x, y, and z coordinates), the velocity of an object by three numbers (the x, y, and z components of its velocity), and its acceleration by three numbers (the x, y, and z components of its acceleration). From the Pythagorean theorem, the magnitudes of these quantities are related by the following expressions…
r^{2} = | x^{2} | + | y^{2} | + | z^{2} |
v^{2} = | v_{x}^{2} | + | v_{y}^{2} | + | v_{z}^{2} |
a^{2} = | a_{x}^{2} | + | a_{y}^{2} | + | a_{z}^{2} |
The relation is more completely described in vector notation where the vectors with a hat (^) over them are unit vectors along the coordinate axes…
r = | x | î | + | y | ĵ | + | z | k̂ |
v = | v_{x} | î | + | v_{y} | ĵ | + | v_{z} | k̂ |
a = | a_{x} | î | + | a_{y} | ĵ | + | a_{z} | k̂ |
Since kinematic vectors at right angles are independent of each other, associated with each direction are functions for displacement, average velocity, and average acceleration…
x = | x(t) | |
v̅_{x} = | Δx | |
Δt | ||
a̅_{x} = | Δv_{x} | |
Δt |
y = | y(t) | |
v̅_{y} = | Δy | |
Δt | ||
a̅_{y} = | Δv_{y} | |
Δt |
z = | z(t) | |
v̅_{z} = | Δz | |
Δt | ||
a̅_{z} = | Δv_{z} | |
Δt |
Similar functions can be derived for instantaneous velocity and instantaneous acceleration…
x = | x(t) | |
v_{x} = | lim | Δx |
Δt → 0 | Δt | |
a_{x} = | lim | Δv_{x} |
Δt → 0 | Δt |
y = | y(t) | |
v_{y} = | lim | Δy |
Δt → 0 | Δt | |
a_{y} = | lim | Δv_{y} |
Δt → 0 | Δt |
z = | z(t) | |
v_{z} = | lim | Δz |
Δt → 0 | Δt | |
a_{z} = | lim | Δv_{z} |
Δt → 0 | Δt |
Or, in the language of calculus…
x = | x(t) | ||
v_{x} = | dx | ||
dt | |||
a_{x} = | dv_{x} | = | d^{2}x |
dt | dt^{2} |
y = | y(t) | ||
v_{y} = | dy | ||
dt | |||
a_{y} = | dv_{y} | = | d^{2}y |
dt | dt^{2} |
z = | z(t) | ||
v_{z} = | dz | ||
dt | |||
a_{z} = | dv_{z} | = | d^{2}z |
dt | dt^{2} |
This application of algebra to geometry is known by several names, analytic geometry being perhaps the most common with coordinate geometry running a close second. Since the person generally credited with the discovery of this subject was the French philosopher and mathematician René Descartes (1596–1650), it is often also called cartesian geometry in his honor. While Descartes was the first to publish his thoughts on this subject, it was almost certainly discovered independently at about the same time by another French mathematician, Pierre de Fermat (1601–1665).
Equations in analytic geometry correspond to curves and surfaces. A typical curve is described by a function that generates a value on the y-axis from every value on the x-axis. A typical surface is described by a function that generates a value on the z-axis from every pair of x-y coordinates. Kinematic equations are described in a way that is somewhat different. The position of a moving object changes with time. Because the x, y, and z values depend on an additional parameter (time) that is not a part of the coordinate system, kinematic equations are also known as parametric equations.
Albert Einstein (1879–1955) turned physics on its head by removing time from the list of parameters and adding it to the list of coordinates. This was a central proposition in his famous theory of relativity; a topic that will be discussed in more detail later in this book. According to Einstein the universe is not three-dimensional with three coordinates — x, y, and z — it's four-dimensional with four coordinates — x, y, z, and t. Objects appear to us to move only because we are swept up in the flow of time. If we could see time the same way we see length, then objects in motion would appear as rigid curves fixed in space — a four-dimensional space that included time, called space-time. In a sense, relativity abolished motion.
Miscellaneous notes
- Instantaneous velocity is always tangential to the curve.
- Acceleration has both tangential and normal/radial/centripetal components.
- A function is a mathematical relation that maps a single output value onto a single input value. Many curves in the cartesian plane can be described by a function y =ƒ(x). Some curves can't. One way around this might be to use an inverse function to draw the curve x =ƒ^{−1}(y). Sometimes even this doesn't work. Try drawing a circle, for instance. The equation r^{2} =x^{2} +y^{2} works but it can't be manipulated into a function. The rearranged equation y = ±√(r^{2} − x^{2}) doesn't cut it. It maps one value of x onto two values of y. I want a honest function, not some mutant equation that would drive me to seek the professional help of a mathematician. I need something so simple a physicist can handle it, or even better, an equation so simple a computer can handle it. One number in, one number out. Well… not exactly one number. What I really want is one number in and one location out, one place on the cartesian plane, one (x, y) coordinate pair. I can do this by parameterizing the curve — turning the x and y coordinates into separate functions of a third variable, the parametric variable, usually identified with the letter t because the best physical meaning to assign to it is time. I don't say "best" cavalierly. I say best because I mean best — best for us, the physicists and those who study physics. Parameterizing a curve in time gives it a physical reality.
- This pair of equations will show you where the object is in space at any moment in time. If I was good, I could rig this computer to show you the way this thing wanders around in a little movie.
- Wow! A movie. Why that's a thousand pictures and a picture's worth a thousand words. That makes a movie worth a million words. Thank you physics man. Now I have a million words — Um — What's a word worth?
- Um. Ida no.
- The simplest parametric equations for an object moving on a circular path at a constant speed are…
x = r cos t y = r sin t - More stuff is needed for this chain of thought.