Physics 1 – Aug 22, 2019 P3 Challenge – Do Now (on slips of paper) Estimate the area (m2) available to each person on earth if there are around 7 ½ billion people and the earth has a radius of around 5000 km. Give a one sigfig order of magnitude estimate. Then describe this area in terms of the images shown in the video. (answers may vary some) Hand in Safety Contract or Syllabus Verifications sheets on front bench. (if still need) Get out Estimation questions and 1.2 WS p1-2 for HMK check

Objectives and Agenda Objectives 1.2 Uncertainties and Errors Assignment: IB 1.2 Uncertainty and Errors Practice Sheet, p 2-4 Agenda Homework Review Standard Deviation practice Types of Uncertainty reporting Error Propagation

Standard Deviation 6) Take the square root of the ans. 𝒙 𝒊 − 𝒙 𝒏−𝟏 𝟐 A better measure of variability in a set of data is the standard deviation. To calculate: 1) Find the average of the data. 𝒙 2) Find the difference between each datum and the average 𝒙 𝒊 − 𝒙 3) Square the differences 𝒙 𝒊 − 𝒙 𝟐 4) Add up all of the squares 𝒙 𝒊 − 𝒙 𝟐 5) Divide by one less than the number of data 𝒙 𝒊 − 𝒙 𝒏−𝟏 𝟐 6) Take the square root of the ans. 𝒙 𝒊 − 𝒙 𝒏−𝟏 𝟐 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.

Find average and standard deviation 85.62 m 85.31 m 83.91 m 84.22 m 85.18 m 84.87 m 85.21 m 84.97 m 84.14 m 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 std listed 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.)

Types of uncertainty reporting Absolute, Fractional and Percent 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: (use the largest) Default based on a single measurement For a small set of data, (max – min)/2 For a large set of data, standard deviation

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 = ∆𝒙 𝒙 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.

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 𝒙 x 100 Ex: 0.000434 * 100 = 0.0434 % (unitless) A favorite of IB especially with MC questions on Paper 1.

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. 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

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. IB formula summary of this rule:

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. IB formula summary of this rule:

Graphing Error Analysis Overview Determine two points on each max/min line generated from the extreme point error boxes Use algebra to determine the slope and intercept of the max and min lines Find the (max – min)/2 for slope to determine the uncertainty in the best fit slope. Find the (max – min)/2 for the intercepts to determine the uncertainty in the best fit intercept. Write summary equation for the best fit line with uncertainties. Determine the percent error, if appropriate. Decide on IV and DV variables. Title the graph and the two axes. Decide on scales for both axes. Plot points. Draw error bars for extreme points (or all) Draw best fit line. Find two points on best fit line. Use algebra to find slope of and intercept of the best fit line. Draw the max and min slope lines https://www.youtube.com/watch?v=Bkp6nHoS_p4 Animation of the process and practice

Exit Slip - Assignment Exit slip – none What’s Due? (Pending assignments to complete.) IB 1.2 Uncertainty and Errors Practice Sheet, p 3-4 except 4 and 9 What’s Next? (How to prepare for the next day) Read IB 1.2 p 11-20