Δs = (v2 + v1) Δt/2;
consequently, the energy of acceleration is
PΔs = w (v2 − v1) (v2 + v1) / 2g = w (v22 − v12) 2g,
(72)
being proportional to the increase in the square of the velocity, and independent of the time.
In order to produce a retardation from the greater velocity v2 to the less velocity v1, it is necessary to apply to the body a resistance connected with the retardation and the time by an equation identical in every respect with equation (71), except by the substitution of a resistance for an effort; and in overcoming that resistance the body performs work to an amount determined by equation (72), putting Rds for Pas.
§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as stored during the acceleration, and restored during the retardation, in a manner analogous to the operation of a reciprocating force (§ 108).
Let there be given the mean velocity V = 1⁄2 (v2 + v1) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v2 − v1, and the extreme velocities v1, v2, which that body must have, in order alternately to store and restore an amount of energy E. By equation (72) we have
E = w (v22 − v12) / 2g
which, being divided by V = 1⁄2(v2 + v1), gives