The Simplifying Consequences of a Favorable Choice of Coordinate System
Suppose a particle moves at a uniform speed, v, toward the northeast beginning at point P at time
t = 0. We write equations describing the particle’s position in a variety of coordinate systems.
(1) We might be tempted to use a stationary coordinate system with
the origin at P and the x- and y-axes in EW and NS orientations.
This perhaps seems simplest, intuitively. In this coordinate system
the solution of the equation of motion, r = vt, is, in terms of its
vector components,
N
y
v
r = (vt cos 45°, vt sin 45°).
45°
P
(2) We can make the expression for r simpler if we rotate
the axes 45° in the counterclockwise direction so that the
x-axis coincides with the direction of v. In this coordinate
system, r has a simpler form,
E
x
NE x
NW y
v
r = (vt, 0).
(3) The ultimate in simplicity is achieved if we take advantage of
the fact that the laws of physics are preserved in any inertial
reference frame, so any such frame, whatever makes things easiest,
is suitable. We choose a frame which is moving uniformly in the
x-direction of the previous frame at speed v as observed in that
previous frame (or in the direction θ = 45° at speed v with respect to
the first frame with the x-direction being the direction of motion).
In this frame the particle remains at the origin, or
P
NW y
NE x
P
 v=0
r = (0, 0).
(4) We conclude with a bad choice. Suppose, in the
coordinate system of choice, that the initial position
N
of the particle (at point P) is given by (x0, y0).
v
Suppose that this coordinate system is moving at
x
y
speed u (with respect to the original frame) in a
φ
direction φ west of north with its x-axis pointed
α
P(x0, y0)
E
towards a direction α counterclockwise from east.
In this coordinate system the position of the particle
u
is given, as usual, by the vector formula
r = r0 + vt, but, in terms of vector components,
since r0 = (x0, y0), and vx = v cos (45° - α) + u sin (φ – α) and vy = v cos (45° + α) + u cos (φ – α),
the expression for r is
r = (x0 + vt cos (45° - α) + ut sin (φ – α), y0 + vt cos (45° + α) - ut cos (φ – α)).