Answer to Question #158549 in Classical Mechanics for Bruce Bond

Question

Four Minute Mile. As part of your training regiment for your eventual pro sports career, you are attempting to complete the famous “4-minute mile”, completing 4 laps of a 400 m track (which is not quite a mile, but it’s close) in a time of 4 minutes or less. As part of your training, you have been running several 3250 m races to improve your sustained long distance speed, and have completed them in an average time of 500 seconds. You have also run several 200 m sprints to improve your sprinting speed, and have completed those sprints in an average time of 25.0 seconds.

a) What are your average sustained long distance speed, and your average sprinting speed? (4)

b) Assuming that you run uniformly at the sustained speed from part a) for the first part of the race, and sprint at the sprinting speed from a) for the final part of the race, at what time should you start sprinting in order to complete the 1.60 km race in 4.00 minutes? How much distance have you covered in this time?

Answer 1

a) The average sustained long distance speed is

$$v=\frac{3250}{500}=6.5\text{ m/s}.$$

The average sprinting speed is

$$u=\frac{200}{25}=8\text{ s}.$$

b) Find the time. Assume L is the total lap length, d is the distance you're sprinting. Therefore, the total time of 4 mins (240 s) can be expressed as:

$$t=\frac{L-d}{v}+\frac du,\\\space\\ 240=\frac{1600-d}{6.5}+\frac{d}{8},\\\space\\ d=213.33\text{ m}.$$

The distance covered is 1600 m, the time when you need to start sprinting is

$$\tau=\frac{L-d}{v}=213.33\text{ s}.$$