Physics

Problem

Two cyclists P and Q start simultaneously from the same lamp post and ride along a straight road towards the east. Cyclist P moves with a constant speed of

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

. Cyclist Q moves with a constant speed of

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

for the first

10\,\text{s}

and then accelerates uniformly so that his speed becomes

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

t=20\,\text{s}

. The lamp post is taken as the origin and the positive

x

-direction is towards the east. a. (i) Write the algebraic expressions for the displacement

x_P(t)

of cyclist P and

x_Q(t)

of cyclist Q in the intervals

0\le t\le 10\,\text{s}

and

10\,\text{s}<t\le 20\,\text{s}

. Indicate clearly the constant acceleration of Q in the second interval. b. (ii) On the same set of axes sketch qualitatively the

x

–

t

graphs of P and Q for

0\le t\le 20\,\text{s}

. Mark the slopes in the two intervals for Q. c. (iii) Using the graphs, compare the instantaneous velocities of P and Q at

t=5\,\text{s}

10\,\text{s}

15\,\text{s}

and

20\,\text{s}

. State which cyclist is faster at each instant. d. (iv) Calculate the numerical value of the distance by which one cyclist is ahead of the other at

t=20\,\text{s}

. State clearly who is ahead.

Intuitive explanation

The key here is tracking position over time for each cyclist separately, then comparing them. Cyclist P is straightforward—constant velocity means constant displacement rate. Cyclist Q has two phases: constant speed for the first 10 seconds, then uniform acceleration for the next 10 seconds. You'll need to find Q's acceleration during phase 2, then use kinematic equations ( $s = ut + \frac{1}{2}at^2$ ) to locate each cyclist at any time $t$ , allowing you to answer questions about their separation or relative motion.

Intuitive explanation

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