Physics Challenges and Homework Review for Uncertainties and Errors

1. Physics 1 – Aug 25, 2017 • P3 Challenge – Do Now (on slips of paper today) ACT Practice papers for “ACT Friday” REAL ACT questions for two passages. It will be timed for 10 min. We will start together. Do not turn over your pages until instructed to do so. Report your answers on P3 papers for me. Keep the questions for reference. Get out 1.2 Worksheet p1 for Homework check.

2. Objectives and Agenda • Homework Review, p1 • IB 1.2 Uncertainties and Errors • Data Analysis (ave and std) • Work with absolute, fractional and percentage uncertainties • Assignment:IB 1.2 Uncertainty and Errors Practice Sheet, p2-4, Extra practice worksheet • Agenda for IB 1.2 Uncertainties and errors • Data Analysis • Types of uncertainties • Error propagation, addition/subtraction • Error propagation, mult./div. • Error propagation, exponents and roots

3. Data Analysis • After you perform experimental trials as outlined by a design, you will have a data set to analyze with several observations of the same event. • Average result • Compare to a known or expected result using a percent error. • A quick estimate of uncertainty can be ½ of range (max – min)/2 • Standard Deviation is used to determine uncertainty. • May use a graph to show relationships between variables.

4. Standard Deviation • A better measure of variability in a set of data is the standard deviation. • To calculate: • 1) Find the average of the data. • 2) Find the difference between each datum and the average • 3) Square the differences • 4) Add up all of the squares • 5) Divide by one less than the number of data • 6) Take the square root of the ans. Not specifically tested on IB, but a “need to know” for the IA. Don’t use the calculator shortcut until you can do it yourself.

5. Find average and standard deviation Possible data set for stopping distances measured by a computer sensor. • Find the average • Find the standard deviation by hand • Check your answer with the stat func 1-var calcs on a graphing calculator (larger sample stdlisted as Sx) • Express your best guess rounded to the proper number of sigfigs with its associated uncertainty. • (Report uncertainty to 1 sigfig, then round value to same decimal place.)

6. Types of uncertainty reporting • Absolute uncertainty – a quantity giving the extremes a measured value falls within • Ex: Absolute uncertainty =∆x • Ex. 23.05 ± 0.01 cm is a best estimate with its absolute uncertainty. • Will have the same unit as x. • Three ways to assess: • Default based on a single measurement • For a small set of data, (max – min)/2 • For a large set of data, standard deviation

7. Types of uncertainty reporting • Fractional uncertainty – the ratio of the absolute uncertainty to the mean value of a quantity. (Sometimes called the relative uncertainty.) • Ex: Fractional uncertainty = • Ex: 0.01 / 23.05 = 0.000434 (unitless) • Because fractional uncertainties are unitless, their relative values can be compared even between quantities that are different.

8. Types of uncertainty reporting • Percent uncertainty – fractional uncertainty x 100%. (Less often used, but helps to build intuition about the meaning of a fractional uncertainty.) • Ex: Percent uncertainty = x 100 • Ex: 0.000434 * 100 = 0.0434 % (unitless)

9. Error propagation – Add/Subtract • When two quantities with uncertainty are added (or subtracted), their absolute uncertainties add. • Even if you subtract, the absolute uncertainties add. • Ex: Q = A + C • Find Q. • Add the uncertainties to find the uncertainty in Q. • Why does this rule work? (Consider Max and Min possible.) • IB formula summary of this rule: If: 𝑦 = 𝑎 ± 𝑏 then: 𝛥𝑦 = 𝛥𝑎+𝛥𝑏 For examples: A =3.5 ± 0.5 B = 0.013 ± 0.001 C = 1.25 ± 0.01 D = 7.1 ± 0.2

10. Error propagation – Mult/Div For examples: A =3.5 ± 0.5 B = 0.013 ± 0.001 C = 1.25 ± 0.01 D = 7.1 ± 0.2 • When two quantities with uncertainty are multiplied (or divided), their fractional uncertainties add. • Ex: Q = BD • Find Q. • Find the fractional uncertainties in B and D. • Add the fractional uncertainties to find the fractional uncertainty in Q. • Multiply Q’s fractional uncertainty by Q to find its ∆Q. • Why does this rule work? (Consider Max and Min possible.) • IB formula summary of this rule:

11. Error propagation – Powers and Roots For examples: A =3.5 ± 0.5 B = 0.013 ± 0.001 C = 1.25 ± 0.01 D = 7.1 ± 0.2 • When two quantities with uncertainty are raised to a power (or rooted), their fractional uncertainties, multiplied by their exponent, add. • Ex: Q = C3 • Find Q. • Find the fractional uncertainties in C. • Multiply the fractional uncertainty by the exponent to find the fractional uncertainty in Q. (For Powers and Roots, sign does not matter.) • Multiply Q’s fractional uncertainty by Q to find its ∆Q. • Why does this rule work? (Consider Max and Min possible.) • IB formula summary of this rule:

12. Exit Slip - Assignment • What’s Due on Tues Aug 25? (Pending assignments to complete.) • IB 1.2 Uncertainty and Errors Practice Sheet, P2-4 • Extra Error Propagation Practice • What’s Next? (How to prepare for the next day) • Read IB 1.2 p 11-19