A particle describes the following motion along a straight track: • From to

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A particle describes the following motion along a straight track: • From

t=0

t=3\,\text{s}

, its velocity increases uniformly from

0

12\,\text{m\,s}^{-1}

. • From

t=3\,\text{s}

t=7\,\text{s}

, its velocity follows the relation

v=12-(t-3)^2

(SI units). • From

t=7\,\text{s}

t=11\,\text{s}

, the velocity decreases uniformly to rest. (i) Write explicit expressions for

v(t)

in the three time intervals stated above. (ii) For each interval, obtain the corresponding expression for instantaneous acceleration and hence determine

a

t=2\,\text{s}

3\,\text{s}

5\,\text{s}

7\,\text{s}

and

9\,\text{s}

. (iii) Comment on the continuity of acceleration at

t=3\,\text{s}

and

t=7\,\text{s}

, supporting your answer with calculations. (iv) Calculate the displacement of the particle during the first

7\,\text{s}

of motion.

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