Mechanics to call the limit of the ratio of a small change in that quantity to the time-interval in which it occurs the rate of change of the quantity. This ratio is the differential coefficient of the quantity with respect to time. Other differential coefficients which occur in Mechanics, in which the independent variable is not the time, are sometimes spoken of as rates, though not frequently. The motion of a point is described when we know not only the path along which it is displaced, but the rates connected with its displacement.
16. Velocity.— The rate of displacement of a point is called its velocity. If the point move in a straight line, and describe equal spaces in any arbitrary equal times, its velocity is constant. A constant velocity is measured by the ratio of the space traversed by the point to the time occupied in traversing that space. If and represent the distances of the point from a fixed point on its path at the instants and , then its velocity is represented by
| (3) |
If the path of the point be curved, or if the spaces described by the point in equal times be not equal, its velocity is variable. The path of a point moving with a variable velocity may be approximately represented by a succession of very small straight lines, which, if the real path be curved, will differ in direction, along which the point moves with constant velocities which may differ in amount. The velocity in any one of these straight lines is represented by the formula . As the interval of time approaches zero, each of the spaces will become indefinitely small, and in the limit the imaginary path will coincide with the real path. The limit of the expression will represent the velocity of the point along the tangent to the path at the time , or, as it is called, the velocity in the path. This limit is usually expressed by
