A marathon runner completes a 42.188km42.188km course in 2h2h , 30 min, and 12s12s . There is an uncertainty of 25m25m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?
Question
A marathon runner completes a 42.188km42.188km course in 2h2h , 30 min, and 12s12s . There is an uncertainty of 25m25m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?
Solution
(a) The percent uncertainty in the distance is calculated by dividing the uncertainty by the total distance and multiplying by 100 to get a percentage. So, (25m / 42188m) * 100 = 0.059%.
(b) The uncertainty in the elapsed time is given as 1 second.
(c) The average speed is calculated by dividing the total distance by the total time. First, we need to convert the time to seconds: 2 hours is 7200 seconds and 30 minutes is 1800 seconds, so the total time is 7200 + 1800 + 12 = 9012 seconds. So, the average speed is 42188m / 9012s = 4.68 m/s.
(d) The uncertainty in the average speed is calculated by adding the relative uncertainties in the distance and time (which are the uncertainties divided by the actual measurements), and multiplying by the average speed. So, the uncertainty in the average speed is (0.00059 + 1/9012) * 4.68 m/s = 0.001 m/s.
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