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 object
involves both translation and rotation. The
motion
of an
aircraft
is particularly complex because the rotations and translations
are coupled together; a rotation affects the magnitude
and direction of the forces which affect translations.
On this page we will consider only the translation of an aircraft
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 moves 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 x-displacement
equals (x1 - x0), the y-displacement equals (y1 - y0), and the z-displacement
equals (z1 - z0). For simplicity on the slide we show only the x coordinate.
d = x1 - x0
The total displacement is a
vector quantity
with the x-, y-, and z-displacements being the
components
of the displacement vector in the coordinate directions. All of the
quantities derived from the displacement are also vector quantities.
The velocity -V of the aircraft through the domain
is the derivative of the displacement with respect to time.
In the X - direction, the average velocity is the displacement divided
by the time interval:
V = (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
where the symbol d / dt is the differential 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 instantaneous velocity V0. Likewise at
the final position x1, y1, z1, and t1,
the velocity could change to
some V1. We are here considering only the x-component of the velocity.
In reality, the aircraft velocity changes in all three directions. Velocity
is a
vector quantity;
it has both a magnitude and a direction.
The acceleration (a) of the aircraft through the domain is the
derivative of the velocity with respect to time.
In the X - direction, the average acceleration is the change in velocity
divided by the time interval:
a = (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 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
velocity of the aircraft as a funtion of the time.
Activities:
Guided Tours
-
Forces, Torques and Motion:
-
Basic Aircraft Motion:
-
Cruising Aircraft:
Navigation ..
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