How fast at exactly 2 s?
"How fast at exactly 2 s?" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "From average speed to instant speed" in Limits and Derivatives. The curve is s = 4.9t² for a dropped stone. The orange chord joins the moment t = 2 s to a second moment, and its slope is the average velocity between them. Shrink the gap h and watch the chord and the average. Each average velocity is the slope of a chord. As the interval shrinks to nothing, the averages close in on one number from both sides and the chord turns into the tangent: the instantaneous velocity is the slope of the tangent, here 19.6 m/s. JEE Main asked 24 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson From average speed to instant speed in Limits and Derivatives. Change the values and watch what happens.
In JEE Main
42 shifts analysed
24 questions from Limits and Derivatives across the 42 JEE Main shifts Lumi has analysed (2017–2026).
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.
Questions
3 answers
- What does the "How fast at exactly 2 s?" simulation show?
- The curve is s = 4.9t² for a dropped stone. The orange chord joins the moment t = 2 s to a second moment, and its slope is the average velocity between them. Shrink the gap h and watch the chord and the average.
- Does JEE Main ask questions from Limits and Derivatives?
- Yes. JEE Main asked 24 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "From average speed to instant speed" in Limits and Derivatives (Class 11 Maths).
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