Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a)  =  +  (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting)

 =  +  (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting)

Let us first define the standard deviation s. Suppose weperform N measurements 1, 2, · · · , N with the average¯. Then the deviation of each measurement is given byδ =  − ¯ with  = 1, 2, · · · , N. The standard deviation s iss =√√√ ∑N=1(δ)2N − 1When we report the average value of N measurements,the uncertainty we should associate with this averagevalue is the standard error.σ = spN =√√√√∑N=1(δ)2N(N − 1)The standard error is smaller than the standard deviationby a factor of 1/pN, since the statistical uncertainty canbe reduced by large number of measurements. Also it isuseful to write σ2 = δ2 ≡ 1N(N − 1)N∑=1δ2 .Suppose we want to determine a quantity  = ƒ (, ),which depends on  and . We want to know the error in = ƒ (, ) if we measure  and  with errors σ and σ .Using the Taylor expansion, we can obtain the law of theerror propagation as follows(δ)2 = ∂ƒ∂2(δ)2 + ∂ƒ∂2(δ)2 + 2 ∂ƒ∂∂ƒ∂(δδ)If the measurements of  and  are uncorrelated, then,on the average, we should expect to find equal distribu-tions of positive and negative values for this term, and weshould expect (δδ) = 0. At the end of the day, using thedefinition of the standard error σ, we can obtainσ =√√√ ∂ƒ∂2σ2 + ∂ƒ∂2σ2Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a)  =  +  (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting).(b)  =  ×  (0.5 pts)(c)  = /  (1 pt)You can find that the relative uncertainty of the prod-uct (quotient) is the root square sum of the individualrelative uncertainties.(d)  = 2 (1 pt)(e)  =  exp(c) with c constant. (0.5 pts)(f)  = 1/ 

For an experiment in this unit, you measure the time of a ball falling from rest as 3.4 ± 0.2 s and you want to calculate the distance the ball has fallen during that time, where you are taking the acceleration due to gravity as a = 9.8 ± 0.1 ms-2. What is the absolute uncertainty (to 1 sig fig) in the distance, calculated from the given values? Question 10) Select one: a. 4 m b. 0.5 m c. 7 m d. 6 m e. 0.3 m

if absolute error in two physical quantities x and y are delta X and delta Y respectively, then absolute error in value of x-y is what

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?