Mathematics

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).

Intuitive explanation

The tangent line to a position–time curve represents the instantaneous velocity at that moment. By finding the slope of the tangent—the rise over run between two points it passes through—you're calculating how fast the trolley is moving at $t=4\,\text{s}$ . This is the core idea of the derivative: the tangent's steepness directly gives you the instantaneous rate of change of position, which is velocity.

Intuitive explanation

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