Sizu Riatto
Calculus · Chapter 10

Position, Speed and Acceleration

Acceleration is the rate of change of speed, the rate of a rate. A positive first derivative with a negative second means rising, but more slowly: the shape behind many headlines.

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Opens on Chapter 10’s example: the library app’s sign-ups, S(t) = 30t² − t³ over 20 days, at day 10.

1. A formula
2. An average acceleration
3. Rising more slowly?
4. Braking and the square of speed
Enter your numbers, then press Work it out.

How it works

Speed is the derivative of position; acceleration is the derivative of speed, so the second derivative of position. A positive second derivative means the graph cups upward (speeding up, or a rise that's gathering pace); a negative one means it frowns (slowing, or a rise that's losing pace). Where the second derivative crosses zero the bending flips: an inflection point, where the speed (or the growth) is at its fastest. For figures taken at even steps, the first differences play the derivative and the differences of those differences play the second derivative. To get an average acceleration in feet per second squared, turn mph into feet a second (× 5,280 ÷ 3,600) and divide by the seconds.

From the book, Chapter 10. 0 to 60 mph in 8 seconds is 11 ft/s every second. A dropped ball accelerates at a steady 32 ft/s². Rises of 3, 2, 1, 0.5 have changes of −1, −1, −0.5: rising, more slowly. The app grows fastest at day 10, 300 sign-ups a day, with 2,000 so far. Jonas accelerates at 2 m/s² at 2 s and 0 at 6 s, when he reaches 9 m/s. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.