Final answer:
The percent uncertainty in the distance of a marathon course is 0.059%. The uncertainty in time is 1 second and does not need a percentage calculation. The average speed is 4.68 m/s, and uncertainty in speed combines the uncertainties of both distance and time.
Step-by-step explanation:
The percent uncertainty in the distance traveled by a marathon runner is calculated by dividing the uncertainty in distance by the actual distance and then multiplying by 100 to get a percentage. To calculate the uncertainty in elapsed time, you simply consider the given uncertainty as it is usually already in a percentage form. The average speed is calculated by dividing the total distance by the total time taken, which must first be converted into seconds. Finally, the uncertainty in the average speed can be calculated by combining the relative uncertainties of distance and time, because when you divide quantities, their relative uncertainties are added together.
(a) Percent uncertainty in distance = (25 m / 42,188 m) × 100 = 0.059%
(b) The uncertainty in the elapsed time is already known to be 1 second.
(c) The average speed (v) in meters per second (m/s) is given by the total distance (d) divided by the total time (t) in seconds. To calculate this, we first convert 2 hours, 30 minutes, and 12 seconds into pure seconds ((2 × 3600) + (30 × 60) + 12 = 9012 seconds), then we find the average speed: v = 42,188 m / 9012 s = 4.68 m/s.
(d) The uncertainty in average speed combines the percent uncertainties of distance and time, considering that the percentage uncertainty for time (1 second out of 9012 seconds) will also be relatively small.