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"Solving" the Motion

Getting back to the subject of Mechanics . . .

One of the reasons the paradigms in the previous chapter emerged was that physicists were always trying to "solve" certain types of "problems" using Newton's SECOND LAW,^12.4

$$F = m \, \ddot{x}$$

This equation can be written

$$\ddot{x} = {1 \over m} \; F$$ (12.1)

to emphasize that it described a relationship between the acceleration  $\ddot{x}$,  the inertial coefficient  m  [usually constant] and the force  F.  It is conventional to call an equation in this form the "equation of motion" governing the problem at hand. When  F  is constant [as for "local" gravity] the "solution" to the equation of motion is the well-known set of equations governing constant acceleration, covered in the chapter on FALLING BODIES. Things are not always that simple, though.

Sometimes the problem is posed in such a way that the force  F  is explicitly a function of time,  F(t). This is not hard to work with, at least in principle, since the equation of motion (1) is then in the form

$$\ddot{x} = {1 \over m} \; F(t)$$ (12.2)

which can be straightforwardly integrated [assuming one knows a function whose time derivative is F(t)] using the formal operation

$$v(t) \; \equiv \; \dot{x} \; \equiv \; \int_0^t \ddot{x} \, dt \; = \; {1 \over m} \int_0^t F(t) \, dt$$ (12.3)

-- which, when multiplied on both sides by  m,  leads to the paradigm of Impulse and Momentum.

In other cases the problem may be posed in such a way that the force  F  is explicitly a function of position,  F(x). Then the equation of motion has the form

$$\ddot{x} = {1 \over m} \; F(x)$$ (12.4)

which can be converted without too much trouble [using the identity   $a \, dx = v \, dv$ ] into the paradigm of Work and Energy.