Velocity at an Instant

Steady velocity is too simple to be very exciting. We now turn to the real problem, the question of variable velocity.

It should be emphasized that the quantity v or s′, for which we are seeking, is intended to measure velocity at an instant. In everyday life we find this quite simple; we glance at the speedometer of a car; the needle points to 60 mph and we conclude that 60 mph is our speed at this instant. But when we start to examine what this means, we meet a certain paradox. The very idea of velocity seems to involve two times, the beginning and end of an interval. We measure velocity in miles an hour, and these words imply that we see how far an object goes in a certain time. If the time allowed is zero, the distance the object goes is zero. However fast it may be going, two photographs of it taken at the same time will show it at the same place.

If in formula (1) we were to try to discover the velocity at an instant, by making a and b coincide, then p and q also would coincide, and the formula would give us 0 ÷ 0 as the velocity—which does not help us at all.

We have ued curves to record the movement of objects. A steep line corresponds to an object moving fast; a gentle slope to an object moving slowly (Figs. 5 and 6). So our question could be posed in terms of curves.