# Speedup vs Slow-down

If you are driving at a speed 60 mph (or kmph, it won’t matter) from one city to another and return at 40 mph, then what would be your average speed?

It’s not 50, it would be 48 (See note 1). A slowdown of 20% cannot be countered with a speed-up of 20%, it takes a speed-up of 33% to counter a slowdown of 20% (See note 2). And that’s the most unintuitive things about slow-downs. In any aspect of life, once a slow-down has occurred, countering it is much harder without taking the risk of serious over-speeding and accidents. The alternative is to accept a lower final average speed.

Note:

- If one-way distance = x, then total time = x/60 + x/40

Average speed = 2x/(x/60 + x/40) = 2 * 60 * 40 / (60 + 40) = 48

The average speed is the Harmonic mean. As a further trivia, the harmonic mean of the two numbers is biased towards the smaller one. The geometric mean is biased towards the larger one (see, that’s why mutual funds love reporting performance in terms of geometric means). The arithmetic mean is dead in the center. - If one-way distance = x, and the desired average speed is v, then time of the first journey with 20% slowdown in x/(4v/5) = 5x/4v

The total time of the journey is 2x/v => time of the return journey = 2x/v – 5x/4v = 3x/4v => average speed for the return journey = 4v/3 = 33% higher than the average speed. - My Physics teacher in grade 9
^{th}taught me this unintuitive slowdown, the result of it mesmerizes me till date.