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Distance-Time Graphs

Two particles...

Question

Two particles A and B start from rest and move for equal time on a straight line. The particle A has an acceleration a for the first half of the total time and 2a for the second half. The particle B has an acceleration 2a for the first and a for the second half. Which particle has covered larger distance?

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Solution

From the velocity–time graph, we have:
Distance covered by A = ( $\frac{1}{2} \times \frac{t}{2} \times \frac{a t}{2} + \frac{1}{2} \times \frac{t}{2} \times \frac{a t}{2} + \frac{t}{2} \times \frac{a t}{2}) = \frac{a t^{2}}{2}$
Distance covered by B = $(\frac{1}{2} \times a t \times \frac{t}{2} + a t \times \frac{t}{2} + \frac{1}{2} \times \frac{a t}{2} \times \frac{t}{2}) = \frac{7}{8} a t^{2}$
∴ Distance covered by A < Distance covered by B

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