A coordinate frame allows us to clearly state and quantify vector qualities like position and velocity. Without them, laws like the above laws of motion are pretty meaningless.
Coordinate frames can have more than three axes and the axes can be in any direction or unit, not just distances or speeds. Some problems in physics, for instance, need coordinate frames that include a time direction, measured in units of seconds. Fortunately, we'll only have to deal with coordinate frames where all three axes define a direction in space. Our vectors, which are defined in relation to the coordinate frames, will have units of displacement (meters), velocity (meters/second) or acceleration (meters/second^2). We've covered velocity. Displacement is the vector describing the relative locations of two objects -- position is the displacement of an object from the origin. An object's acceleration describes how it's velocity changes over time -- the "time rate of change of velocity".
Notice the pattern between displacement, velocity and acceleration. Each is a property of an object. Each successive property has the same units as the previous, but divided by seconds. That's because each successive property is the time rate of change of the previous one. Velocity describes the way displacement changes with time and acceleration describes the way velocity changes with time. All three are vectors. The first element (direction) of the velocity vector describes the time rate of change of the first element (direction) of the position vector. The second element of the velocity vector describes the time rate of change of the second element of the position vector. Velocity and acceleration relate in the same way.
As a result, if we know the velocity of an object, we can predict how it's position will change with time and if we know it's acceleration, we can predict how it's velocity will change with time. To predict how an object will move, then, we can use the second law to find it's acceleration, which, along with a knowing the object's position and velocity at sometime in the past, we can use to describe how an object moves.
Now we just need to know what force and mass mean, and what exactly that third law is saying. We'll cover those next time.
Thanks to Jorge Stolfi for created in the coordinate frame image and to Wikimedia for hosting it.
Saturday, March 19, 2011
XA-092
I'm considering using this blog to host a sort of Introduction To Aerospace (Engineering) class. I think this will benefit me in a variety of ways: brushing me up on the basics of my field, helping me polish my rather tarnished writing skills, improve my equally tarnished pedagogical abilities, motivate me to build up a library of basic aerospace tools and possibly give me the opportunity to make promising contacts in the community and help me scout out promising candidates for any future "aerospace drafts."
I'm planning on making the class as accessible as possible will still honestly covering the field. I figure assuming my readers have some knowledge of algebra is a good start. So I'm viewing this as "Aerospace (Engineering) for 9th graders." Were it a college class, it might be "Remedial Aerospace." I'm also going to rely on graphics as little as possible. I'll use them, but I'll do my best to explain all of the concepts adequately in the text. I'll likely include optional "advanced" content as well. Advanced posts will include Python and Haskell code -- Python, because it's so accessible and powerful and Haskell, because I think it has a lot of potential as a language for engineers.
If this test post works out well, I'll post a link to it and any successor posts on aerospace.reddit.com. (My inspiration for this coming from this link.)If the Intro class works out, I'll continue trying to cover the field in greater and greater detail. I hope to have the first experimental post done sometime tonight. I'll be asking for feedback on the experimental post amongst my aerospace, engineering, teaching and other friends... if you fall into one of those categories, please feel free to speak up whenever.
Laws of Motion, Coordinate Systems, Vectors
Aerospace (engineering) may be viewed as the "art and science of making things move through air and space". To practice this art, we have to first understand how things move. It turns out that there's a simple set of physics laws -- rules the describe the physical world -- that quite accurately describe how all sorts of everyday objects move. These are called "Newton's laws of motion."
Historically, there are three of them. Roughly stated, they are:
1. An object in motion keeps moving the same way unless it's acted upon by a force. An object at rest stays at rest unless acted on by an outside force.
2. Force equals mass times acceleration.
3. Every applied force has an equal and opposite applied force.
Today, we usually only consider the latter two laws as modern scientists note that these laws apply, that is, are true, only in what's called an "inertial frame of reference." The first indirectly describes such a frame. Inertial frames of reference (alternatively, "Inertial reference frames") are "coordinate frames" that do not rotate or change velocity.
Velocity has a specific meaning -- it's the speed of an object and it's direction. It can be written, for instance, as "5 kilometers per hour, due east." Since space is three dimensional, velocity can also be written as an object's speed in each dimension, for example "3 kph right, 2.5 kph up, -1.2 kph out."
The left-right, down-up and in-out directions are given short hand names in math and physics, usually x, y and z. Each direction has it's own axis, or line running in that direction. All three axes (the plural for axis) intersect each other at one point called the origin. Each axis shares a name with it's corresponding direction: "x axis, y axis, z axis". Altogether, the axis, the origin and the space they sit in make up a "coordinate frame of reference," or coordinate frame for short. Inertial frames of reference are a special case of such frames. We call the individual values, 3 kph, etc, "coordinates."
An object's position in space can be described the same way. "2 meters right, -10 meters up, 0 meters out." We call position, velocity and other properties that can be describe with magnitudes and directions "vectors." We often write vectors as three numbers inside parentheses. (2 meters right, -10 meters up, 0 meters out). We can set the order we mention the directions by convention, so we can write (2, -10, 0 ) meters and our meaning is understood to be the same as above. Furthermore, the directions don't have to be left-right, down-up and in-out, they can be tilted as well. Lastly, we can represent an unknown vector using variables. In the figure below uppercase X,Y and Z are the axis/direction names and x,y and z are variables representing the coordinates.
A coordinate frame allows us to clearly state and quantify vector qualities like position and velocity. Without them, laws like the above laws of motion are pretty meaningless.
Coordinate frames can have more than three axes and the axes can be in any direction or unit, not just distances or speeds. Some problems in physics, for instance, need coordinate frames that include a time direction, measured in units of seconds. Fortunately, we'll only have to deal with coordinate frames where all three axes define a direction in space. Our vectors, which are defined in relation to the coordinate frames, will have units of displacement (meters), velocity (meters/second) or acceleration (meters/second^2). We've covered velocity. Displacement is the vector describing the relative locations of two objects -- position is the displacement of an object from the origin. An object's acceleration describes how it's velocity changes over time -- the "time rate of change of velocity".
Notice the pattern between displacement, velocity and acceleration. Each is a property of an object. Each successive property has the same units as the previous, but divided by seconds. That's because each successive property is the time rate of change of the previous one. Velocity describes the way displacement changes with time and acceleration describes the way velocity changes with time. All three are vectors. The first element (direction) of the velocity vector describes the time rate of change of the first element (direction) of the position vector. The second element of the velocity vector describes the time rate of change of the second element of the position vector. Velocity and acceleration relate in the same way.
As a result, if we know the velocity of an object, we can predict how it's position will change with time and if we know it's acceleration, we can predict how it's velocity will change with time. To predict how an object will move, then, we can use the second law to find it's acceleration, which, along with a knowing the object's position and velocity at sometime in the past, we can use to describe how an object moves.
Now we just need to know what force and mass mean, and what exactly that third law is saying. We'll cover those next time.
Thanks to Jorge Stolfi for created in the coordinate frame image and to Wikimedia for hosting it.
A coordinate frame allows us to clearly state and quantify vector qualities like position and velocity. Without them, laws like the above laws of motion are pretty meaningless.
Coordinate frames can have more than three axes and the axes can be in any direction or unit, not just distances or speeds. Some problems in physics, for instance, need coordinate frames that include a time direction, measured in units of seconds. Fortunately, we'll only have to deal with coordinate frames where all three axes define a direction in space. Our vectors, which are defined in relation to the coordinate frames, will have units of displacement (meters), velocity (meters/second) or acceleration (meters/second^2). We've covered velocity. Displacement is the vector describing the relative locations of two objects -- position is the displacement of an object from the origin. An object's acceleration describes how it's velocity changes over time -- the "time rate of change of velocity".
Notice the pattern between displacement, velocity and acceleration. Each is a property of an object. Each successive property has the same units as the previous, but divided by seconds. That's because each successive property is the time rate of change of the previous one. Velocity describes the way displacement changes with time and acceleration describes the way velocity changes with time. All three are vectors. The first element (direction) of the velocity vector describes the time rate of change of the first element (direction) of the position vector. The second element of the velocity vector describes the time rate of change of the second element of the position vector. Velocity and acceleration relate in the same way.
As a result, if we know the velocity of an object, we can predict how it's position will change with time and if we know it's acceleration, we can predict how it's velocity will change with time. To predict how an object will move, then, we can use the second law to find it's acceleration, which, along with a knowing the object's position and velocity at sometime in the past, we can use to describe how an object moves.
Now we just need to know what force and mass mean, and what exactly that third law is saying. We'll cover those next time.
Thanks to Jorge Stolfi for created in the coordinate frame image and to Wikimedia for hosting it.
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