A particle is in rectilinear motion when it always lies in the same straight line, and when its distance from a point in that line varies with the time. It moves with uniform speed if it passes over equal distances in equal intervals of time, whatever their length. The speed is represented by a positive number, and is measured by the distance passed over in a unit of a time. The velocity of a particle is the directed speed with which it moves, and is positive or negative according to the direction of the motion. Hence in uniform motion the velocity is given by the equation
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| (1.1) |
Since s may be positive or negative, v may be positive or negative, and the speed is the numerical value of v. The same value of v is obtained whatever interval of time is taken so long as the corresponding value of s is used.
The speed and velocity are variable when the particle does not describe equal distances in equal times ; and it is necessary to define in this case what is meant by the speed and velocity at an instant. Suppose a particle passes over the distance Δs in the time Δt and suppose the interval of time Δt approaches the limit zero in such a manner that it always contains the instant t. Suppose, further, that for every Δt the corresponding Δs is taken. Then the velocity at the instant t is defined as
 | (1.2) |
and the speed is the numerical value of 
Uniform and variable velocity may be defined analytically in the following manner. The distance s counted from a fixed point, is considered as a function of the time, and may be written
Then the velocity may be defined by the equation
where φ′(t) is the derivative of φ(t) with respect to t. The velocity is said to be constant, or uniform, if φ′(t) does not vary with t or, in other words, if φ(t) involves linearly in the form φ(t) = at + b where a and b are constants. It is said to be variable if the value of φ′(t) changes with t.
Some agreement must be made to denote the direction of motion. An arbitrary point on the line may be taken as the origin and the distances to the right counted as positive, and those to the left, negative. With this convention, if the value of s determining the position of the body increases algebraically with the time the velocity will be taken positive ; if the value of s decreases as the time increases the velocity will be taken negative. Then, when is given refin magnitude and sign, the speed and direction of motion are determined.