Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Coordinates and Components
'Cause we're all
Bold as love,
Just ask the axis. How do we convert these ideas into mathematics? The figure below shows a good way of connecting the intuitive ideas to the numbers. In one dimension, we impose a number line with an x coordinate on a certain stretch of space. In two dimensions, we imagine a grid of squares which we label with x and y values, as shown in figure b.
But of course motion doesn't really occur in a series of discrete hops like in chess or checkers. Figure c/ shows a way of conceptualizing the smooth variation of the x and y coordinates. The ball's shadow on the wall moves along a line, and we describe its position with a single coordinate, y, its height above the floor. The wall shadow has a constant acceleration of 9.8 m/s^{2}. A shadow on the floor, made by a second light source, also moves along a line, and we describe its motion with an x coordinate, measured from the wall.
The velocity of the floor shadow is referred to as the x component of the velocity, written v_{x}. Similarly we can notate the acceleration of the floor shadow as a_{x}. Since v_{x} is constant, a_{x} is zero. Similarly, the velocity of the wall shadow is called v_{y}, its acceleration a_{y}. This example has a_{y} = 9.8 m/s^{2}. Because the earth's gravitational force on the ball is acting along the y axis, we say that the force has a negative y component, F_{y}, but F_{x} = F_{z} = 0. The general idea is that we imagine two observers, each of whom perceives the entire universe as if it was flattened down to a single line. The yobserver, for instance, perceives y, v_{y}, and a_{y}, and will infer that there is a force, F_{y}, acting downward on the ball. That is, a y component means the aspect of a physical phenomenon, such as velocity, acceleration, or force, that is observable to someone who can only see motion along the y axis. All of this can easily be generalized to three dimensions. In the example above, there could be a zobserver who only sees motion toward or away from the back wall of the room.
Projectiles move along parabolas.
What type of mathematical curve does a projectile follow through space? To find out, we must relate x to y, eliminating t. The reasoning is very similar to that used in the example above. Arbitrarily choosing x = y = t = 0 to be at the top of the arc, we conveniently have x = Δx, y = Δy, and t = Δt, so
We solve the second equation for t = x/v_{x} and eliminate t in the first equation:
Since everything in this equation is a constant except for x and y, we conclude that y is proportional to the square of x. As you may or may not recall from a math class, y x^{2} describes a parabola. → Solved problem: A cannon page 191, problem 5
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