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1. IF a point change its position in space it is said to


2. All motion has reference to space and time, and since a point may, under different circumstances, pass over different spaces in equal intervals of time, or require different intervals of time to pass over equal spaces, the mind necessarily conceives the idea of quickness or slowness of motion. The degree of this quickness or slowness is called velocity.

3. If the moving point pass over equal spaces in equal successive intervals of time, its velocity is said to be uniform. It is evident that the velocity of a moving point will be greater or less in exact proportion as the space it passes over in any given time is greater or less, or as the time required for the point to pass over any given space is less or greater; so that the measure of the velocity varies as the space passed over when the time is constant, and inversely as the time when the space is constant, i.e. if v be the measure of the velocity with which a moving point describes a space s in a time t, voc v x = .


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4. If, with the unit of velocity, a space o be described in T, we shall have

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As σ and are perfectly arbitrary, we may assume any values we please for them. The simplest assumption is that

7=1, σ =1, and then v=3.

If t=1, v = s.

5. The assumptions we have made determine the unit of velocity to be that with which the unit of space is described in the unit of time; and the measure of any velocity to be the space passed over in an unit of time.

The units of space and time are perfectly arbitrary.

6. Since the simplicity of an investigation depends greatly upon the choice of units, it may often be advantageous to obtain an expression for a velocity in terms of other units of space and time than those in terms of which it is already expressed.

Thus, suppose we wish to obtain the measure of a velocity in terms of a new set of units of space and time, a and b times respectively as great as the original units. If v be the measure of the velocity referred to the original units, the moving point would describe the space v in the unit of time, and therefore it would describe the space bv in the new unit of time, which is b times as great as the former; therefore, as the new unit of space is a times as great as the former, the number of such units



described will be -; this, therefore, is the measure of the velocity sought.

E. g. If the velocity of a moving point be measured by 20 when a foot and a second are the units of space and time, what will be its measure when a yard and a minute are the units?

The point would move over 20 feet in 1′′,

and therefore over 20 × 60 feet in 60" or 1',

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therefore the measure of the velocity is 400.

In common language, the rate of 20 feet per second is the same as that of 400 yards per minute.

7. If s be the space passed over in time t by a point moving uniformly with a velocity v, then will svt. For v units of space are passed over in the unit of time, and therefore vt units of space are passed over in the time t, i. e. s = vt.

If a be the initial distance of the moving point from a fixed point in its path, and s the distance at time t, then s=a+vt, provided that the point be moving away from the fixed point. If however it be moving towards it, s will


This shews that if we write v for v, we reverse the direction of the motion; or, in other words, if velocities in any one direction be considered positive, those in the opposite direction must be considered negative.

We may obviously represent a velocity by a straight line drawn in the direction of the motion, and equal in length to the magnitude of the measure of the velocity, so that if a point were to move uniformly along the line, it would arrive at the extremity of it at the end of an unit of time.

8. The Parallelogram of Velocities.

"If two adjacent sides of a parallelogram represent in magnitude and direction two velocities simultaneously existing in the motion of a point, the resulting velocity will be represented in magnitude and direction by the diagonal drawn through the point of intersection of those two sides."

In order to conceive the motion more clearly, let the point be supposed to be compelled to move uniformly along the line AB, while AB is transferred so as to be always parallel to itself, its extremity moving uniformly along AC.

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