Lesson 1 of 12 · 12 min

From average speed to instant speed

NCERT §12.1, §12.2, Tables 12.1 to 12.3

Meera's science club is at a cliff-top viewpoint above a river gorge. They drop a pebble from the edge and a tablet on a tripod films it, logging how far it has fallen at each moment. How fast is the pebble moving exactly 2 seconds after release?

The story this chapter follows: The science club at the cliff-top viewpoint

Imagine Meera's science club at a viewpoint on top of a 44.1 m cliff above a river gorge. A pebble dropped from the edge falls s = 4.9t² metres in t seconds and splashes into the river at t = 3 s. A tablet logs the fall, a trail runs up to the edge, the deck is a curved arc with a rail, and a rope swing hangs nearby. Every limit and derivative in this chapter is measured there.

The lesson in notes

In short

Calculus studies how the value of a function changes as the input changes. This chapter first builds the idea of a derivative from motion, then defines a limit, and then returns to define the derivative properly and find it for standard functions.

A body dropped from a tall cliff falls s = 4.9t² metres in t seconds. At t = 1, 1.5, 2, 2.5 and 3 s it has fallen 4.9, 11.025, 19.6, 30.625 and 44.1 m.

Average velocity is distance covered divided by time taken. Over the first 2 s it is 19.6/2 = 9.8 m/s; from 1 s to 2 s it is (19.6 − 4.9)/1 = 14.7 m/s.

Shrinking the interval so that it ends at 2 s gives 17.15 (from 1.5 s), 18.62 (from 1.8 s), 19.11 (from 1.9 s), 19.355 (from 1.95 s) and 19.551 m/s (from 1.99 s): the averages rise.

Intervals that start at 2 s give 29.4 (to 4 s), 24.5 (to 3 s), 22.05, 20.58, 20.09, 19.845 and 19.649 m/s (to 2.01 s): these fall.

Both runs close in on one value, so the velocity at the instant t = 2 s lies between 19.551 and 19.649 m/s. That value, the instantaneous velocity, is the rate of change of distance at t = 2.

On the distance-time graph each average velocity is the slope of a chord through the point at t = 2. As the other end slides in, the chord turns into the tangent at that point, so the instantaneous velocity is the slope of the tangent.