Mathematics

Problem

For a body moving along a straight line, the displacement as a function of time is given by

x(t) = 3t^{2} - 12t + 5

where

x

is in metres and

t

in seconds, for the time interval

0 \le t \le 6\,\text{s}

. a. (i) Derive an expression for the instantaneous velocity

v(t)

as a function of time. b. (ii) Calculate the displacement at

t = 1\,\text{s}

and

t = 5\,\text{s}

, and hence determine the average velocity between these two instants. c. (iii) Find the instantaneous velocity at

t = 3\,\text{s}

(the mid-point of the interval considered in part (ii)). d. (iv) Examine whether the instantaneous velocity at the mid-point equals the average velocity over the interval

1\,\text{s}

5\,\text{s}

, and justify your conclusion mathematically.

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