U. K. XV.-2 but one, taken in the same order round, are jointly equivalent to a velocity represented by that one side, taken in the opposite order; also that a point which has simultaneously. velocities represented by the successive sides of any polygon, taken all in the same order round, is at rest. The second law of motion (see iNloTioN, LAWS OF) enables us to inter pret this geometrical theorem into the physical truths known as the triangle and polygon of forces in statics.
Rate of change of velocity is called acceleration. It is measured in the same way as velocity itself. Thus if the change take place in the direction of motion it affects merely the amount, not the direction of the velocity; and an acceleration a adds (or subtracts, if it be negative) a feet per second front the velocity affected. Thus it is found that gravity produces an acceleration of about 32.2 on all falling bodies; so that if a stone be let fall, its velocity after t seconds is 32.21. If it be thrown down with a velocity v, its velocity in t seconds is v +32.21. If thrown upward with the same velocity, iu seconds its velocity becomes v 32.21, so that it will stop and begin to descend after 11 seconds have elapsed.
32.2 The space passed over by the stone in t seconds is easily calculated by the help of i the average velocity. For since in any of the above cases the velocity increases (or diminishes) uniformly, its average value during any interval is the average of its values at the beginning and end of the interval. Hence for the stone simply let fall: Initial velocity = 0, Velocity after t seconds = 32.21, Average velocity during the first t seconds = 16.11. Hence, space described in t seconds = t x average velocity = So that the spaces described are as the squares of the times.
But if the acceleration be not in the direction of motion, the direction and magnitude of the velocity will generally change.
To exhibit this geometrically, sir W. Rowan Hamilton (q.v.) invented the following beautiful construction of what he called the hodograph of the motion. Let 0 be any fixed point, and from it draw lines OP, OQ, etc., representing at every instant in direction and magnitude the velocity of the moving point. The extremities of such lines will form a curve, such as PQ in the figure. If OP and OQ be any two of these, the change of velocity is represented (as above) by the third side, PQ, of the triangle. As Q is taken nearer and nearer to I', PQ becomes more and more nearly the tangent to the hodograph, so that the tangent at P has the direction of the accel eration, and the rate at which P moves round the hodograph is the magnitude of the acceleration.
If we consider any uniform motion, we see that the hodograph is a circle (its radius being the magnitude of the velocity), and from this it is easy to see that in uniform motion the acceleration is always perpendicular to the direction of motion. If we consider uniform motion, with velocity V, in a circle of radius It, the hodograph at once shows 2 that the acceleration is It ' and is directed toward the center of the circle.
Translated into physics, acceleration (multiplied by the mass of the moving body) is the measure of the force which acts on the body. So the above simple example shows that, to keep a mass moving uniformly in a circle, it must be drawn toward the center by a force proportional directly to the square of the velocity, and inversely to the radius. This is the physical explanation of the so-called centrifugal force (see CENTRAL FoRcEs).