Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

from P). Remembering that the velocity is tangential, it is therefore completely determined by the speed v (the path and position
P being known).
The term velocity as commonly used means either speed or
velocity, according to the context. For motion along a given path
the "compound" of two speeds is their ordinary sum, but velocities
derived from displacements combine by the parallelogram law of
displacements, and not by numerical addition, since they are displacements per unit time.
ACCELERATION-THE HODOGRAPH
The average acceleration from time t to time t' is the average
increase of velocity per unit time.
The acceleration at time t is the limit of the above average
acceleration.
From a fixed point 0 (fig. 1) draw lines OB, OH, OH', etc.,
representing, in direction and magnitude, the velocities of the
point at A, P, P', etc., (and so parallel to the tangents to the path
at those points, and of lengths representing the speeds at those
points, V, v, v', etc.) The curve B..H.. H'..., is called the
hodograph of the given motion.
Now HH' is the resultant of the velocities OH' and -OH i. e.,
the increase of velocity from time t to time t'; and if it be produced
to T' in the ratio 1: (t'-t), HT' will represent the above average
acceleration; and it will also represent the average velocity of the
corresponding point in the hodograph, between the same times.
Thus its limit HT, on the tangent to the hodograph at H, represents simultaneously the acceleration at P, and the velocity at H.
In words,
The acceleration of a moving point is the velocity of the corresponding point in its hodograph.
TANGENTIAL AND NORMAL ACCELERATIONS
In fig. 1, draw the perpendicular, TK, to OH produced. Then
HK and KT are called the tangential and normal accelerations, at
P, because they are the components of the whole acceleration HT,
parallel and perpendicular to the tangent at P. The normal ac'celeration KT, drawn from P, must lie towards the center of curvature, C, since it is in the direction in which the velocity PS begins
to turn as the moving point leaves P, or towards the concave side,of the path.
We proceed to find the measures of the tangential and normal
accelerations.
Lay off on the velocities OB,...OH,...OH'..., their corre

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