Final answer:
The percent uncertainty in the marathon's distance is approximately 0.059%. The uncertainty in elapsed time is 1 second. The average speed of the runner is about 4.68 meters per second, and the uncertainty in average speed would be a combination of the uncertainties in distance and time.
Step-by-step explanation:
To calculate the percent uncertainty in distance, we divide the uncertainty by the total distance covered and then multiply by 100. For a marathon runner who completes a 42.188-km course with an uncertainty of 25 m, the calculation would be as follows:
(a) Percent uncertainty in distance = (Uncertainty / Distance) × 100% = (25 m / 42,188 m) × 100% ≈ 0.059%.
(b) To calculate the uncertainty in the elapsed time, we simply consider the uncertainty in seconds since the measurement is already in seconds:
Uncertainty in elapsed time = 1 s.
The total time in seconds for the marathon is 2 hours, 30 minutes, and 12 seconds, which is converted to 2 × 3600 + 30 × 60 + 12 = 9012 s. The average speed is then:
(c) Average speed = Distance / Time = 42,188 m / 9012 s ≈ 4.68 m/s.
Uncertainty in average speed can be more complex to calculate, as it involves derivatives or approximation techniques. Alternatively, for a simple estimate, one could propagate the relative uncertainties of the distance and time taken:
(d) Relative uncertainty in speed = Relative uncertainty in distance + Relative uncertainty in time.
Assuming small percentages, the uncertainty in average speed (in m/s) would be close to the sum of the individual percent uncertainties since the speed is directly proportional to both distance and time.