Question

A runner times his speed around a circular track with a circumference of \(0.478 \mathrm{mi} .\) At the start he is running toward the east and the track starts bending toward the north. If he goes halfway around, he will be running toward the west. He finds that he has run a distance of \(0.750 \mathrm{mi}\) in 4.00 min. What is his (a) average speed and (b) average velocity in \(\mathrm{m} / \mathrm{s} ?\)

Short answer

Question: A runner takes 4.00 minutes to run around half of a circular track with a circumference of 0.478 miles. Given that he runs a total distance of 0.750 miles, calculate (a) his average speed in meters per second (m/s), and (b) his average velocity in m/s. Answer: (a) The runner's average speed is \(5.03\,\mathrm{m/s}\). (b) The runner's average velocity is \(1.02\,\mathrm{m/s}\).

Step-by-step solution

Find the Conversion Factor (from miles to meters)

In order to convert the distance and circumference from miles into meters, find out how many meters are in 1 mile. 1 mile = 1609.34 meters.

Convert Distance and Circumference to Meters

Now, use the conversion factor to convert the given distance traveled, 0.750 miles, and track circumference, 0.478 miles, into meters. Distance in meters: \(0.750 \mathrm{mi} * 1609.34 \mathrm{m/mi} = 1207 \mathrm{m}\) (rounded to nearest meter) Circumference in meters: \(0.478 \mathrm{mi} * 1609.34 \mathrm{m/mi} = 769 \mathrm{m}\) (rounded to nearest meter)

Calculate Displacement

Because the runner goes halfway around the circular track, the displacement is a straight line from start to end (known as the diameter of the circle). To find the displacement, use the circumference as follows: Diameter (displacement) = Circumference / π Displacement = \(769\textrm{m} / \pi \approx 245\textrm{m}\) (rounded to nearest meter)

Convert Time to Seconds

The given time is 4.00 minutes. To convert this into seconds, multiply the time by 60: Time in seconds: \(4.00 \textrm{min} * 60 \textrm{s/min} = 240\textrm{s}\)

Calculate Average Speed

Now that we have the distance in meters and time in seconds, we can find the average speed using the formula: average speed = distance traveled/time taken. Average speed = \(\frac{1207\mathrm{m}}{240\mathrm{s}} = 5.03\,\mathrm{m/s}\)

Calculate Average Velocity

To find the average velocity, use the formula: average velocity = displacement/time taken Average velocity = \(\frac{245\mathrm{m}}{240\mathrm{s}} = 1.02\,\mathrm{m/s}\) So, the runner's (a) average speed is \(5.03\,\mathrm{m/s}\) and (b) average velocity is \(1.02\,\mathrm{m/s}\).