Applications of the Calculus to Mechanics

Cover Applications of the Calculus to Mechanics
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A n = (ds/dC) (dr/dC) where T = arctan v y /v x .
20. The differential equations of plane motion. The differential equations of plane motion were expressed in the last paragraph in the relation between force and acceleration. Usually they are written =' The problem of determining the " complete description " from these equations together with proper auxiliary conditions is one whose difficulty depends entirely upon the nature of the force. The simplest case is that in which the component /. , .
... Depends only upon x and dx/dt, and f y only upon y and dy/dt. In this case the solution of each differential equation is a separate problem. Usually, however, both f x and / depend upon both x and y and upon v x and v y . The equations are therefore simultaneous differ- ential equations, and are solved by various devices depending upon the special problems attacked, important among which is the elimination of one of the dependent variables.
If we know the path in advance (see footnote, p. 55), we only need determine the function s = (t) giving the way in which the path is described.


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