3.3: Speed and acceleration

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Problem 84 Jack and Jill went up the hill, and averaged 2 mph on the way up. They then turned round and went straight back down by the same route, this time averaging 4 mph. What was their average speed for the round trip (up and down)?

Problem 85

(a) (i) A cycling road race requires one to complete 3 laps of a long road circuit. On the first lap I average 40 km/h; on the second lap I average 30 km/h; and on the third lap I only average 20 km/h. What is my average speed for the whole race?

(ii) I cycle for 3 hours round the track of a velodrome, averaging 40 km/h for the first hour, 30 km/h for the second hour, and 20 km/h for the final hour. What is my average speed over the whole 3 hours?

(b) Two cyclists compete in an endurance event.

(i) The first cyclist pedals at 60 km/h for half the time and then at 40 km/h for the other half. The second cyclist pedals at 60 km/h for half of the total distance and then at 40 km/h for the remaining half. Who wins?

(ii) In a two hour event, the first cyclist pedals at u km/h for the first hour and then at v km/h for the second hour. The second cyclist pedals at u km/h for half of the total distance and then at v km/h for the remaining half. Who wins?

* the arithmetic mean

$\begin{matrix} \frac{u + v}{2} \end{matrix}$

of two positive quantities u, v, and

* the harmonic mean

$\begin{matrix} \frac{2}{\frac{1}{u} + \frac{1}{v}} . \end{matrix}$

(ii) Give a purely algebraic proof of your inequality in (i).

Problem 86 A train started from a station and, moving with a constant acceleration, covered a distance of 4 km, finally reaching a speed of 72 km/hour. Find the acceleration of the train, and the time taken for the 4 km.

Problem 87 (Average speed of an accelerating car) A typical car (and maybe also a typical train!) does not move with constant acceleration. Starting from a standstill, a car moves through the gears and “accelerates more quickly” in lower gears, when travelling at lower speeds, than it does in higher gears, when travelling at higher speeds. Use this empirical fact to prove that the average speed of a car accelerating from rest is more than half of its final measured speed after the acceleration.

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