A trolley rolls down an inclined track

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

A trolley rolls down an inclined track. Its position

s

is measured at regular intervals and plotted, giving a curved position–time graph. At

t=4\,\text{s}

a tangent is drawn to the curve. This tangent is found to pass through the points (

2.5\,\text{s},\,6\,\text{m}

) and (

5.5\,\text{s},\,42\,\text{m}

). Additional measurements show that the trolley is at the origin at

t=0\,\text{s}

and at

s=16\,\text{m}

t=4\,\text{s}

. Answer the following. (a) Part 1: (i) Explain why the curvature of the position–time graph indicates that the trolley is moving with non-uniform speed. (b) Part 2: (ii) Using the given points on the tangent, calculate the instantaneous speed of the trolley at

t=4\,\text{s}

. (c) Part 3: (iii) Determine the average speed of the trolley between

t=0\,\text{s}

and

t=4\,\text{s}

, and compare it with the instantaneous speed obtained in part (ii).

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