iSoul In the beginning is reality

Displacement vs. arc length

As pointed out here, average speed does not equal the magnitude of average velocity. But the instantaneous speed does equal the magnitude of instantaneous velocity. For example, the average velocity of one orbit is zero but the average speed is positive.

Consider a section of a curve as below:

The arc length of this section of the curve is Δs. The displacement is Δr. This with the horizontal and vertical differences Δx and Δy makes a triangle. The Pythagorean theorem gives the hypotenuse of the triangle:

(Δr)² = (Δx)² + (Δy)².

Clearly Δs is longer than Δr since the shortest path between two points is a straight line. Now consider a differential version of this:

Then (dr)² = (dx)² + (dy)². But in this case ds is not shorter than dr since they are both differentials, that is, infinitesimals.

So the differential ratios with denominator Δt as Δt → 0 equal dr/dt and ds/dt, which are equal. The instantaneous speed uses either the differential displacement or the differential arc length.

But determination of the arc length is much more difficult than determination of the hypotenuse. So the instantaneous speed, velocity, and acceleration are defined in terms of the hypotenuse, not the arc length. After all, arc length can be determined by integrating the differential hypotenuse.

So the displacement is used instead of the arc length.

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