If a bicyclist rides at 10 miles per hour for 10 minutes, and then speeds up to 20 miles per hour for the next 20 minutes, what is her average speed for this ride?

Many people's first instinct is to simply average the two speeds, but this common mistake overlooks a key component: time. The average speed of a journey is always the total distance traveled divided by the total time it took. Because the bicyclist in this scenario spent more time traveling at the faster speed, that portion of the ride has a greater influence on the final average. This principle is a fundamental concept in physics and is crucial for accurately planning travel and understanding motion.

To find the correct average speed, we first need to determine the total distance covered. In the first 10 minutes (or 1/6 of an hour), riding at 10 miles per hour, the cyclist travels 1 and 2/3 miles (10 mph * 1/6 h). In the next 20 minutes (or 1/3 of an hour) at 20 miles per hour, she covers a much greater distance of 6 and 2/3 miles (20 mph * 1/3 h). Adding these distances together gives a total journey of 8 and 1/3 miles.

The total time for the ride is 30 minutes, or half an hour. By dividing the total distance of 8 and 1/3 miles by the total time of 1/2 an hour, we arrive at the correct average speed. This calculation reveals that the cyclist's average speed for the entire trip was 16 and 2/3 miles per hour, a figure weighted more heavily towards the faster speed at which she spent more time.