among force, mass, and acceleration, but they stop short of discussing velocity, time, and distance. These are covered here. In the interest of simplicity, we assume here that acceleration is constant. Then,
where
Δ (cap delta) means “change in”
Vf = final velocity at time tf
Vi = initial velocity at time ti
If we start the time at ti = 0 and rearrange the above, then
If we start the time at ti = 0 and Vi = 0 (brakes locked before takeoff roll) and rearrange the above where Vf can be any velocity given, for example liftoff velocity, then
The distance s traveled in a certain time is
where the average velocity Vav is
And incorporating Eq. 1.4, and substituting for Vf, we get
which yields
Solving Eqs. 1.4 and 1.5 simultaneously and eliminating t, we can derive a third equation:
Equations 1.3–1.6 are useful in calculating takeoff and landing factors, and are studied in more detail in Chapters 8 and 9.
EXAMPLE
An aircraft that weighs 15 000 lb begins from a brakes‐locked position on the runway, and then accelerates down the runway with a net force of 5000 lb until liftoff at a velocity of 110 kts. Calculate the average acceleration down the runway, the average time it takes to reach liftoff speed, and the total takeoff distance on the runway.
First, to calculate the acceleration, we need find the force (F) and the mass of the aircraft during the takeoff roll, Eq. 1.3: F = m a
Finding the mass:
Finding the average acceleration:
Average time to liftoff:
Total takeoff distance:
ROTATIONAL MOTION
Without derivation, some of the relationships among tangential (tip) velocity, Vt; radius of rotation, r; revolutions per minute, rpm; centripetal forces, CF; weight of rotating parts, W; and acceleration of gravity, g, are shown below. A more detailed discussion regarding rotorcraft can be found in Chapter 15 of this textbook.
(1.7)
(1.8)
(1.9)
For our discussion, the units of work will be measured in ft‐lb.
ENERGY AND WORK
Energy is the ability to do work. In physics, work has a meaning different from the popular definition. You can push against a solid wall until you are exhausted but, unless the wall moves, you are not doing any work. Work requires that a force must move an object (displacement) in the direction of the force. Another way of saying this is that only the component of the force in the direction of movement does any work:
There are many kinds of energy: solar, chemical, heat, nuclear, and others. The type of energy that is of interest to us in aviation is mechanical energy.
There are two kinds of mechanical energy: The first is called potential energy of position, or more simply potential energy, PE. No movement is involved in calculating PE. A good example of this kind of energy is water stored behind a dam. If released, the water would be able to do work, such as running a generator. As a fighter aircraft zooms to a zenith point, it builds PE; once it starts to accelerate downward, it converts PE to KE. PE equals the weight, W, of an object multiplied by the height, h, of the object above some base plane:
(1.10)
The second kind of mechanical energy is called kinetic energy, KE. As the name implies, kinetic energy requires movement of an object. It is a function of the mass, m, of the object and its velocity, V:
(1.11)
The total mechanical energy, TE, of an object is the sum of its PE and KE:
(1.12)
The law of conservation of energy states that
