Applications of the derivative in mechanics

Included also are applications to velocity and rectilinear motion.

Consider the motion of a point P on the straight line AB.

Figure 6.12: Scan of Granville's graphic of the rectilinear motion.

Let $s$ be the distance measured from some fixed point as A to any position of P, and let $t$ be the corresponding elapsed time. To each value of $t$ corresponds a position of P and therefore a distance (or space) $s$. Hence $s$ will be a function of $t$, and we may write

$$s = f(t)$$

Now let $t$ take on an increment $\Delta t$; then $s$ takes on an increment^6.9 $\Delta s$, and

$$\frac{\Delta s}{\Delta t} = {\rm the\ average\ velocity}$$ (6.20)

of P during the time interval $\Delta t$. If P moves with uniform motion, the above ratio will have the same value for every interval of time and is the velocity at any instant.

For the general case of any kind of motion, uniform or not, we define the velocity (or, time rate of change of s) at any instant as the limit of the ratio $\frac{\Delta s}{\Delta t}$ as $\Delta t$ approaches the limit zero; that is,

$$v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t},$$

or

$$v = \frac{ds}{dt}$$ (6.21)

The velocity is the derivative of the distance (= space) with respect to the time.

To show that this agrees with the conception we already have of velocity, let us find the velocity of a falling body at the end of two seconds.

By experiment it has been found that a body falling freely from rest in a vacuum near the earth's surface follows approximately the law

$$s = 16.1t^2$$ (6.22)

where $s$ = space fallen in feet, $t$ = time in seconds. Apply the General Rule, §4.7, to (6.22).

FIRST STEP. $s + \Delta s = 16.1(t + \Delta t)^2 = 16.1 t^2 + 32.2 t \cdot \Delta t + 16.1(\Delta t)^2$.

SECOND STEP. $\Delta s = 32.2t \cdot \Delta t + 16.1(\Delta t)^2$.

THIRD STEP. $\frac{\Delta s}{\Delta t} = 32.2t + 16.1\Delta t =$ average velocity throughout the time interval $\Delta t$.

Placing $t = 2$,

$$\frac{\Delta s}{\Delta t} = 64.4 + 16.1\Delta t$$ (6.23)

which equals the average velocity throughout the time interval $\Delta t$ after two seconds of falling. Our notion of velocity tells us at once that (6.23) does not give us the actual velocity at the end of two seconds; for even if we take $\Delta t$ very small, say $\frac{1}{100}$ or $\frac{1}{1000}$ of a second, (6.23) still gives only the average velocity during the corresponding small interval of time. But what we do mean by the velocity at the end of two seconds is the limit of the average velocity when $\Delta t$ diminishes towards zero; that is, the velocity at the end of two seconds is from (6.23), $64.4$ ft. per second.

Thus even the everyday notion of velocity which we get from experience involves the idea of a limit, or in our notation

$$v = \lim_{\Delta t \to 0} \left ( \frac{\Delta s}{\Delta t} \right ) = 64.4 \ ft./sec.$$

The above example illustrates well the notion of a limiting value. The student should be impressed with the idea that a limiting value is a definite, fixed value, not something that is only approximated. Observe that it does not make any difference how small $16.1 \Delta t$ may be taken; it is only the limiting value of $64.4 + 16.1\Delta t$, when $\Delta t$ diminishes towards zero, that is of importance, and that value is exactly $64.4$.