We live in world that is defined by three spatial dimensions and one
time dimension. Objects can move within this domain in two ways.
An object can
translate,
or change location, from one
point to another.
And an object can
rotate,
or change its attitude.
In general, the motion of an
aircraft
involves both translation and rotation.
This makes the analysis of aircraft motion much more complex than
ordinary objects.
On this page we will consider only the translation of an object
within our domain. We can specify the location of our aircraft at
any time (t) by specifying three coordinates (x, y, and z) on an
orthogonal coordinate system. An orthogonal coordinate system has each
of its coordinate directions perpendicular to all other coordinate directions.
Initially, our aircraft is at point "0", with coordinates x0, y0, and z0
at time t0. In general, the aircraft can move through the domain until
at some later time t1 the aircraft is at point "1" with coordinates
x1, y1, and z1. We can specify the displacement (d) in each coordinate direction
by the difference in coordinate from point "0" to point "1". The xdisplacement
equals (x1  x0), the ydisplacement equals (y1  y0), and the zdisplacement
equals (z1  z0). For simplicity on the slide we show only the x coordinate.
d = x1  x0
The average velocity (V) of the aircraft through the domain is the displacement
divided by the time difference. In the X  direction, the average velocity
is (x1  x0) / (t1  t0). This is only an average velocity; the aircraft
could speed up and slow down inside the domain. At any instant, the aircraft
could have a velocity that is different than the average. If we shrink the
time difference down to a very small (differential) size, we can define the
instantaneous velocity to be the differential change in position divided by the
differential change in time (V = dx / dt) from calculus. So when we initially
specified the location of our aircraft with (x0, y0, z0, and t0) coordinates,
we could also specify an initial V0 instantaneous velocity. Likewise at
the final position (x1, y1, z1, and t1), the velocity could change to
some V1. (We are here considering only the xcomponent of the velocity. In
reality, the aircraft velocity can change in all three directions. Velocity
is a vector quantity; it has both a magnitude and a direction associated with it.)
V = (x1  x0) / (t1  t0)
The average acceleration (a) of the aircraft through the domain is the
difference in the velocity
divided by the time difference. In the X  direction, the average acceleration
is (V1  V0) / (t1  t0). As with the velocity, this is only an average
acceleration. At any instant, the aircraft
could have an acceleration that is different than the average. If we shrink the
time difference down to a very small (differential) size, we can define the
instantaneous acceleration to be the differential change in velocity divided by the
differential change in time (a = dV / dt) from calculus. From Newton's
second law
of motion, we know that forces on an object produce accelerations. If we can
determine the forces on an aircraft, and how they change, we can use the
equations presented on this slide to determine the location and
translation of the aircraft.
a = (V1  V0) / (t1  t0)
Navigation..
 ReLiving the Wright Way
 Beginner's Guide to Aeronautics
 NASA Home Page
 http://www.nasa.gov
