Question

A runner times his speed around a track with a circumference of $0.50 \mathrm{mi} .\( He finds that he has run a distance of \)1.00 \mathrm{mi}$ in 4.0 min. What is his (a) average speed and (b) average velocity magnitude in \(\mathrm{m} / \mathrm{s}\) ?

Short answer

Answer: The runner's (a) average speed is 6.71 m/s, and (b) his average velocity magnitude is 0 m/s.

Step-by-step solution

Convert given units to meters and seconds

We need to convert the given distances from miles to meters and the time from minutes to seconds. We do that by multiplying by the conversion factor. 1 mile = 1609.34 meters 1 minute = 60 seconds Circumference = 0.50 mi * 1609.34 m/mi = 804.67 m Distance traveled = 1.00 mi * 1609.34 m/mi = 1609.34 m Time taken = 4.0 min * 60 s/min = 240 s Now we have all distances in meters and times in seconds.

Calculate average speed

Average speed is calculated by dividing the distance run by the time taken. We have the distance traveled (1609.34 m) and the time taken (240 s): Average_speed = (Distance_traveled) / (Time_taken) Average_speed = 1609.34 m / 240 s ≈ 6.71 m/s The runner's average speed is approximately 6.71 m/s.

Calculate the total displacement

Displacement means the change in position from the starting point. Since the runner is moving in a circular track, after covering the 1-mile distance, he has done 2 full laps around the track (1 mile / 0.5-mile circumference). Returning to the starting point after each lap, the total displacement becomes 0. Total_displacement = 0 m

Calculate the average velocity magnitude

Since the total displacement is 0, the average velocity magnitude is also 0: Average_velocity_magnitude = (Total_displacement) / (Time_taken) Average_velocity_magnitude = 0 m / 240 s = 0 m/s The runner's average velocity magnitude is 0 m/s. In conclusion, the runner's (a) average speed is 6.71 m/s, and (b) his average velocity magnitude is 0 m/s.