Theory question 107
DifferentiationWhen a stone is thrown vertically upwards, its distance d metres after t seconds is given by the formula d = 60t − 10t2. Draw the graph of d = 60t − 10t2 for values of t from 1 to 5 seconds using 2cm to 1 unit on the t axis and 2cm to 20 units on the d axis. (a) Using your graph (i) how long does it take to reach a height of 70 metres? (ii) determine the height of the stone after 5 seconds (iii) after how many seconds does it reach its maximum height? (b) Determine the slope of the graph when t = 4 seconds.
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Table: t = 1, 2, 3, 4, 5 gives d = 50, 80, 90, 80, 50. Plot and join with a smooth curve.
(a)(i) d = 70 at t ≈ 1.6 s (on the way up; again at t ≈ 4.4 s on the way down).
(ii) d = 50 m at t = 5 s.
(iii) Maximum height (90 m) at t = 3 s.
(b) Draw the tangent at t = 4; its gradient ≈ −20 (m/s) — the stone is falling.
The slope of a distance–time graph is the velocity; after the top it is negative.