Coordinates and Components

'Cause we're all Bold as love, Just ask the axis.
Jimi Hendrix

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.

b / This object experiences a force that pulls it down toward the bottom of the page. In each equal time interval, it moves three units to the right. At the same time, its vertical motion is making a simple pattern of +1, 0, -1, -2, -3, -4, . . . units. Its motion can be described by an x coordinate that has zero acceleration and a y coordinate with constant acceleration. The arrows labeled x and y serve to explain that we are defining increas- ing x to the right and increasing y as upward.

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/s². 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.

c / The shadow on the wall shows the ball's y motion, the shadow on the floor its x motion.

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/s².

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 y-observer, 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 z-observer who only sees motion toward or away from the back wall of the room.

A car going over a cliff.

Projectiles move along parabolas.

e / A parabola can be defined as the shape made by cutting a cone parallel to its side. A parabola is also the graph of an equation of the form y x2.

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 x² describes a parabola.

→ Solved problem: A cannon page 191, problem 5

f / Each water droplet follows a parabola. The faster drops' parabolas are bigger.

Discussion Questions

A

At the beginning of this section I represented the motion of a projectile on graph paper, breaking its motion into equal time intervals. Suppose instead that there is no force on the object at all. It obeys Newton's first law and continues without changing its state of motion. What would the corresponding graph-paper diagram look like? If the time interval represented by each arrow was 1 second, how would you relate the graph-paper diagram to the velocity components vx and vy?

B

Make up several different coordinate systems oriented in different ways, and describe the ax and ay of a falling object in each one.