### Velocity and Acceleration

VELOCITY AND ACCELERATION

Motion is a progressive change of position of a body. Velocity is the rate of motion, that is, the rate of change of position. When the velocity of a body is the same at every moment during which the motion takes place, the latter is called uniform motion. When the velocity is variable and constantly increasing, the rate at which it changes is called acceleration; that is, acceleration is the rate at which the velocity of a body changes in a unit of time, as the change in feet per second, in one second. When the motion is decreasing instead of increasing, it is called retarded motion, and the rate at which the motion is retarded is frequently called the deceleration. If the acceleration is uniform, the motion is called uniformly accelerated motion. An example of such motion is found in that of falling bodies.

Newton's Laws of Motion.—The first clear statement of the fundamental relations existing between force and motion was made in the seventeenth century by Sir Isaac Newton, the English mathematician and physicist. It was put in the form of three laws, which are given as originally stated by Newton:
1) Every body continues in its state of rest, or uniform motion in a straight line, except in so far as it may be compelled by force to change that state.
2) Change of motion is proportional to the force applied and takes place in the direction in which that force acts.
3) To every action there is always an equal reaction; or, the mutual actions of two bodies are always equal and oppositely directed.

Motion with Constant Velocity.—In the formulas that follow, S = distance moved; V = velocity; t = time of motion, θ = angle of rotation, and ω = angular velocity; the usual units for these quantities are, respectively, feet, feet per second, seconds, radians, and radians per second. Any other consistent set of units may be employed.
Constant Linear Velocity:
S = V × t V = S ÷ t t = S ÷ V

Constant Angular Velocity:
θ = ωt ω = θ ÷ t t = θ ÷ ω

Relation between Angular Motion and Linear Motion: The relation between the angular velocity of a rotating body and the linear velocity of a point at a distance r feet from the center of rotation is:
V(ft per sec) = r(ft) × ω(radians per sec)

Similarly, the distance moved by the point during rotation through angle θ is:

Linear Motion with Constant Acceleration.—The relations between distance, velocity, and time for linear motion with constant or uniform acceleration are given by the formulas in the accompanying Table 1. In these formulas, the acceleration is assumed to be in the same direction as the initial velocity; hence, if the acceleration in a particular problem should happen to be in a direction opposite that of the initial velocity, then a should be replaced by − a. Thus, for example, the formula Vf = Vo + at becomes Vf = Vo − at when a

and Vo are opposite in direction.