Simulation · Maths · Class 11

How fast at exactly 2 s?

From the lesson From average speed to instant speed in Limits and Derivatives. Change the values and watch what happens.

The idea behind it

NCERT §12.1, §12.2, Tables 12.1 to 12.3

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.

1 more