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Basic Math for Physics

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Basic Math for Physics

  1. 1. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 1 Rapid Learning Center Chemistry :: Biology :: Physics :: Math Rapid Learning Center Presents …Rapid Learning Center Presents … Teach Yourself AP Physics in 24 Hours 1/67 *AP is a registered trademark of the College Board, which does not endorse, nor is affiliated in any way with the Rapid Learning courses. Basic Math foras c at o Physics Physics Rapid Learning Series Rapid Learning Center www.RapidLearningCenter.com/ © Rapid Learning Inc. All rights reserved. Wayne Huang, Ph.D. Keith Duda, M.Ed. Peddi Prasad, Ph.D. Gary Zhou, Ph.D. Michelle Wedemeyer, Ph.D. Sarah Hedges, Ph.D.
  2. 2. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 2 Learning Objectives Using algebra to solve for a i bl By completing this tutorial, you will learn math skills commonly used in physics: variable Performing calculations with significant figures Using scientific notation Performing calculations with exponents & scientific 3/67 with exponents & scientific notation Trigonometry Using the quadratic formula Calculator tips Concept Map experimental science Physics is an experimental science calculator tips Follow calculator tips Requires Previous content New content ExperimentationExperimentation Conclusions Results & Conclusions Data taken with correct # of sig figs Data taken with correct # of sig figs CalculationsCalculations Follow rules with sig figs Follow rules for calculations with sig figs Leads to Often requires During which When doing, Be sure to May be written in Depend on 4/67 QuadraticQuadratic Formula TrigonometryTrigonometryAlgebra Rules for exponents Rules for working with exponents Use scientificUse scientific notation Might use When using, follow
  3. 3. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 3 Algebra Algebra is often used when sol ing ph sics problems 5/67 solving physics problems. Algebra Rules If a number is being … to an Then … that number on both Undo by doing the opposite to both sides: Example unknown variable sides Added Subtract Example 5 = x + 2 -2 -2 5-2 = x 6/67 Subtracted Add 3 = x – 6 +6 +6 3+6 = x
  4. 4. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 4 Algebra Rules-2 If a number is being … to an Then … that number on both Undo by doing the opposite to both sides: Example 2 = 4x 4 4 2÷4 = x unknown variable sides Multiplied Divide Example 7/67 Divided Multiply 2* 6 = x *2 2 2*6 = x Example: Velocity Velocity problems require algebraic work: displacement(d) Velocity time(t) = Example: Solve for distance The unknown variable (d) is being divided by 25 s time(t) 25s d m/s52 = * 25 s25 s * 8/67 The unknown variable (d) is being divided by 25 s. Multiply both sides by 25 s dm/s52*25s = d1300m =
  5. 5. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 5 Example: Force Net force problems also require algebraic work: frictionappliednet FFF += frictionF52N25N += Example: Solve for force of friction The unknown variable (Ff i ti ) is being -52 N-52 N 9/67 The unknown variable (Ffriction) is being added to 52 N Subtract 52 N from both sides frictionF52N25N =− frictionF27N =− Example: Gravitational Force When solving for a variable on the bottom, you need to move it to the top first: Example: Solve for d ( )22118 g)(50kg)(50k /kgNm106 6N101 7 −− ×=× Move d to top ( ) 2 d /kgNm106.6N101.7 ×=× ( ) 2 22118 d g)(50kg)(50k /kgNm106.6N101.7 −− ×=×d2 * * d2 ( ) ( ) g)(50kg)(50k/kgNm106.6N101.7d 221182 −− ×=× Solve for d 10/67 ( ) ( ) 1.7x10-8 N 1.7x10-8 N N)10(1.7 kg))(50kg)(50/kgNm10(6.6 d 8 2211 − − × × = 3.11md =
  6. 6. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 6 Calculations & Significant Figures This section describes the role 11/67 of significant figures in calculations. Calculations, Significant Figures Usage Data is taken very carefully to the correct number of significant figuressignificant figures. 12/67 When performing calculations, you cannot end up with greater precision than what you actually measured in the lab!
  7. 7. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 7 Always Round Last Always complete all calculations first… 13/67 And then round as a final step. Rounding: Adding & Subtracting Adding & subtracting: Perform the calculation1 Example: Perform the following operation: 10.027 g 1 5 g 3 decimal places 1 decimal place Determine the least # of decimal places in the problem Round the answer to that many decimal places. 2 3 14/67 - 1.5 g 8.527 g 1 decimal place 8.5 g 1 Least = 1 2 3
  8. 8. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 8 Rounding: Multiplying & Dividing Multiplying & Dividing: Perform the calculation1 Example: Perform the following operation: Determine the least # of significant figures in the problem Round the answer to that many significant figures. 2 3 1 10.027 m 6 68467 m/s 5 sig figures = 15/67 6.68 m/sLeast = 3 2 3 1.50 s 6.68467 m/s 3 sig figures = Scientific Notation Scientific notation is often used th h t th t d f h i 16/67 throughout the study of physics.
  9. 9. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 9 Definition – Scientific Notation Scientific Notation - A short-hand method of writing very large or very ll bsmall numbers. Scientific Notation uses powers of 10. 17/67 Writing in Scientific Notation Rules for writing scientific notation: Decimal is moved to follow the 1st non-zero number Number of times the decimal is moved = power of 10 1 2 Examples: Write in scientific notation: 1027500.456 g 1 Number of times the decimal is moved = power of 10 “Big” (>1) numbers positive powers “Small” (<1) numbers negative powers 2 3 123456 18/67 1.027500456 g The decimal point was moved 6 times 1.027500456 The original number was “big” so it will be a +6 power 1.027500456 x 106 g 1 2 3 123456
  10. 10. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 10 Writing in Scientific Notation 2 Rules for writing scientific notation: Decimal is moved to follow the 1st non-zero number Number of times the decimal is moved = power of 10 1 2 Examples: Write in scientific notation: 0.0007543 g 1 Number of times the decimal is moved = power of 10 “Big” (>1) numbers positive powers “Small” (<1) numbers negative powers 2 3 1234 19/67 7.543 g The decimal point was moved 4 times 0007.543 The original number was “small” so it will be a -4 power 7.543 x 10-4 g 1 2 3 1234 Sig Figures & Scientific Notation Scientific Notation can also be used to write a specific number of significant figures. Example: Your calculator gives you: 10200 g And you need to write the result with 4 significant figures 10200 g 3 significant figures 10200. g 5 significant figures 4321 20/67 1.020 X 104 g 4 significant figures! Use scientific notation 4321
  11. 11. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 11 Reading Scientific Notation - 1 Rules for reading scientific notation: Power of 10 = number of times to move the decimal P iti “Bi ” ( 1) b 1 Examples: Write out the following number 1 Positive powers “Big” (>1) numbers Negative powers “Small” (<1) numbers 2 3.25 X 10-6 g00000 123456 . 21/67 1 2 The decimal point will move 6 times The power is negative, so we’ll make the number “small”—move the decimal forward (left) 0.00000325 g 123456 Reading Scientific Notation - 2 Rules for reading scientific notation: Power of 10 = number of times to move the decimal P iti “Bi ” ( 1) b 1 Examples: Write out the following number 1 Positive powers “Big” (>1) numbers Negative powers “Small” (<1) powers 2 7.2004 X 103 mL. 123 22/67 1 2 The decimal point will move 3 times The power is positive, so we’ll make the number “big”—move the decimal point right (backwards) 7200.4 mL 123
  12. 12. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 12 Exponents 23/67 Exponent Use Exponents are used in unit conversions and when using scientific notation. There are several rules that govern 24/67 g calculations with exponents.
  13. 13. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 13 Exponent Rule #1 “Rule of 1” A b t th f 1 th i i l b1 Example: Any number to the power of 1 = the original number1 251 = 25 25/67 Exponent Rule #2 “Rule of 0” A b t th f 0 12 Example: Any number to the power of 0 = 12 160 = 1 26/67
  14. 14. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 14 Exponent Rule #3 “Product rule” Wh lti l i dd th t3 Example: When multiplying, add the exponents.3 112*114 = 11(2+4) This rule does not apply when the bases = 11(6) 27/67 pp y are different! Example: 112*124 (11*12)(2+4)= Exponent Rule #4 “Quotient rule” Wh di idi bt t th t4 Example: When dividing, subtract the exponents.4 = 15(8-3) This rule does not apply when the bases = 15(5) 158 153 28/67 pp y are different! Example: 158 123 (15 12) (8-3) =
  15. 15. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 15 Exponent Rule #5 “Power rule” When there is an exponent on an exponent multiply5 Example: When there is an exponent on an exponent, multiply the two exponents together. 5 (82)3 = 8(2*3) = 8(6) 29/67 (8 ) = 8( ) = 8( ) Exponent Rule #6 “Negative rule” When a number with a negative exponent is placed on6 Example: When a number with a negative exponent is placed on the opposite side of the fraction, the exponent is made positive. 6 9-2 = 1 92 30/67 9 = 92
  16. 16. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 16 Summary of Exponent Rules A b t th f 0 1 Any number to the power of 1 = the original number. 251 = 25 1 Exponent on an exponent: multiply the two exponents5 Dividing: subtract the exponents. 32÷38 = 3-6 4 Multiplying: add the exponents. 32*38 = 310 3 Any number to the power of 0 = 1 250 = 12 31/67 Negative exponent: place on the opposite side of the fraction and the exponent is made positive 3-2 = 1/32 6 together. (32)3 = 36 5 Calculations with Scientific Notation There are severale e a e se e a similar rules when working with scientific notation calculations. Scientific Notation 6.02x1023 32/67
  17. 17. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 17 Sci. Notation Calculation Rule #1 Addition Add th b d k th f 101 Example: Add the numbers and keep the same power of 101 6x104 + 2x104 = (6+2)x104 This rule does not apply when the powers = 8x104 33/67 pp y p of 10 are different! Example: 6x103 + 6x106 (6+6)x10(3+6)= Sci. Notation Calculation Rule #2 Subtraction S bt t th b d k th f 102 Example: Subtract the numbers and keep the same power of 102 3x102 - 1x102 = (3-1)x102 This rule does not apply when the powers = 2x102 34/67 pp y p of 10 are different! Example: 3x102 - 6x106 (3-6)x10(2-6)=
  18. 18. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 18 Sci. Notation Calculation Rule #3 Multiplication M lti l th b d dd th f 103 Example: Multiply the numbers and add the powers of 10.3 3x104 * 2x106 = (3*2)x10(4+6) = 6x1010 35/67 Sci. Notation Calculation Rule #4 Division Di id th b d bt t th f 104 Example: Divide the numbers and subtract the powers of 10.4 8x102 ÷ 2x106 = (8 ÷ 2)x10(2-6) = 4x10-4 36/67
  19. 19. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 19 Sci. Notation Calculation Rule #5 Taking a scientific notation number to a power T k th b t th d lti l th5 Example: Take the number to the power and multiply the power by the power of 10. 5 (3x102)3 = (3)3x10(2*3) = 27x106 37/67 Sci. Notation Calculation Rule #6 Taking a root of a scientific notation number T k th t f th b d di id th f6 Example: Take the root of the number and divide the power of 10 by the root. 6 = =√3x102 √3x10(2/2) √3x10(1) 38/67
  20. 20. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 20 Summary of Sci. Notation Rules Subtraction: Subtract the numbers and keep the same2 Addition: Add the numbers and keep the same power of 10 2X103 + 3X103 = 5X103 1 Division: Divide the numbers and subtract the powers of 10 2X104 / 3X103 = 0.67X101 4 Multiplication: Multiply the numbers and add the powers of 10 2X106 * 3X103 = 6X109 3 Subtraction: Subtract the numbers and keep the same power of 10 2X103 - 3X103 = -1X103 2 39/67 Roots: Take the root of the number and divide the power of 10 by the root √(3x102)= √3 X 101 6 Powers: Take the number to the power and multiply the power by the power of 10 (2X103)3 = 8X109 5 Geometry & Trigonometry Both aspects of math are frequently used in physics 40/67 frequently used in physics.
  21. 21. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 21 Definition – Trigononmetric Functions Trigonometric Functions – Functions of an angle that give ratios of theof an angle that give ratios of the sides of a triangle within that right triangle. 41/67 Trigonometric Functions The three common trig functions work with right triangles. itTrig functions use one of the hypotenuse opposite sinθ = hypotenuse adjacent cosθ = Trig functions use one of the non-90° angles and discuss the sides of the triangle as they relate to that angle tetoθ 42/67 adjacent opposite tanθ = Right angle θ“Theta”—angle in question Opposit Adjacent to θ
  22. 22. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 22 Remembering the Functions S in Opposite Make sure you Hypotenuse C os Adjacent Hypotenuse Make sure you pronounce it with a long “O” in “SOH” so that you don’t think it’s an “A”! 43/67 Hypotenuse T an Opposite Adjacent Example: Solve for the length of the hypotenuse Trigonometry Example 25° 5.2 m You know an angle and a length opposite the angle and you want to know the hypotenuse. Opposite & hypotenuse are used in “sin” 5 2 44/67 hypotenuse 5.2m sin25 =Hypotenuse * * Hypotenuse sin 25° sin 25° sin25 5.2m hypotenuse = hypotenuse=12m
  23. 23. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 23 Solving for an Angle If you know the two sides of the triangle, use the “arc” trig function to solve for the angle The “arc” is symbolized with ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − hypotenuse opposite sinθ 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − hypotenuse adjacent cosθ 1 y “-1” exponent and is found as a “2nd” function on your calculator tetoθ 45/67 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − adjacent opposite tanθ 1 Right angle θ “Theta θ” - angle in question Opposit Adjacent to θ Example: Solve for the unknown angle Trigonometry Example - Solving for Angle ? You know the sides opposite and adjacent to an unknown angle. “tan-1” can be used 3.7 m 7.5 m 46/67 3.7m 7.5m tanθ 1− = 63.7θ =
  24. 24. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 24 Definition - Vector Vectors – Quantities with a magnitude and a direction. Trigonometry is used when calculating with vectors 47/67 calculating with vectors.. Quadratic Equation The next topic will be the d ti ti 48/67 quadratic equation.
  25. 25. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 25 Definition – Quadratic Formula Quadratic Formula – n. formula used to find “x” in an equation that contains “x” and “x2” . 4acbb x 2 −±− = 49/67 2a 0cbxaxwhere 2 =++ Example When working velocity problems, you may end up with equations that contain an “x” and an “x2 Example: Solve for x (“x” represents “time”) 2a 4acbb x 2 −±− = 085x1x2 =−+ 1*2 8)*1*(455 x 2 −−±− = From above: a = 1, b = 5, c = -8 50/67 12 2 575 x ±− = Using the + give x = 1.27 Using the – gives x = -6.27 A negative answer isn’t possible for time
  26. 26. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 26 Graphing Graphing is commonly used in 51/67 physics. Graphing and Slopes Graphing is used extensively in physics to visualize and analyze data. Slopes can give meaningful information. Y-axis Position Velocity X-axis Time Time Slope Velocity Acceleration 52/67 In Calculus, these are called derivatives…Velocity is the derivative of position.
  27. 27. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 27 Finding Average Slopes Use the final and initial conditions to find the average slope. 80 0 10 20 30 40 50 60 70 53/67 0 0 2 4 6 8 10 Initial conditions: x = 0 and y = 3 Final conditions: x = 8 and y = 67 12 12 xx yy slope − − = 8 08 367 slope = − − = Average slope = 8 Finding Instantaneous Slopes Use tangents to find the slope at an instant on a curve Find slope at x = 5 10 20 30 40 50 60 70 80 54/67 0 10 0 2 4 6 8 10 Initial conditions: x = 5 and y = 28 Final conditions: x = 7.7 and y = 53 12 12 xx yy slope − − = 9.3 57.7 2853 slope = − − = Instantaneous slope at x = 5 is 9.3
  28. 28. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 28 35 Graphing Direct Relationships Many relationships in physics are direct 0 5 10 15 20 25 30 0 2 4 6 8 10 12 Position Slope is a straight line 55/67 Time Equation of line is in format: y = mx + b where: m = slope b = y-intercept Graphing Quadratic Relationships Some relationships in physics are quadratic 80 0 10 20 30 40 50 60 70 56/67 0 0 2 4 6 8 10 Equation of line is in format: y = kx2 + b where: k = a constant b = y-intercept
  29. 29. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 29 Graphing Inverse Relationships Some relationships in physics are inverse 2 9 3 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 57/67 0 1 2 3 4 5 6 7 8 Equation of line is in format: y = k (1/x) where: k = a constant Calculator Survival Finally, we will discuss important i f i l l 58/67 tips for using your calculator correctly.
  30. 30. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 30 Calculator Survival Many times a student has a problem set up correctly, but makes a mistake entering it into the calculator. 59/67 The following slides will show how to avoid common calculator mistakes. Calculator Tip #1 Dividing by more than one number L)atm)(2.5(1.25 K)L)(273atm)(1.0(1.00 T2 =Example: Common mistake: Typing in * tells the calculator that the number following it is on top of the expression. 1.00 * 1.0 * 273 ÷ 1.25 * 2.5Entering in: 60/67 Correct method: 1.00 * 1.0 * 273 ÷ 1.25 ÷ 2.5Entering in: Always use ÷ to designate a number on the bottom.
  31. 31. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 31 Calculator Tip #2 Using scientific notation 2 3 10*9.8 10*1.5 D =Example: Common mistake: 1.5 * 10 ^ 3 ÷ 9.8 * 10 ^ 2Entering in: Typing in * tells the calculator that the number following it is on top of the expression. 61/67 Correct method: 1.5 EE 3 ÷ 9.8 EE 2Entering in: Always use EE (or EXP) key to enter in scientific notation. Calculator Tip #3 Adding/subtracting & multiplying/dividing together 100* 1.37 1.371.25 error% − =Example: Common mistake: 1.25 - 1.37 ÷ 1.37 * 100Entering in: Always use parenthesis around addition Addition and subtraction must happen first…before multiplication and division. 62/67 Correct method: (1.25 - 1.37) ÷ 1.37 * 100Entering in: Always use parenthesis around addition and subtraction when combining with other operations.
  32. 32. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 32 Calculator Tip #4 Using powers with negative numbers 2 10−Example: Common mistake: (-10)2Entering in: To make something negative keep the Anything inside the parenthesis will be squared as well. 63/67 Correct method: -(102)Entering in: To make something negative, keep the negative sign out of the exponent operation. Or -102 Summary of Calculator Tips S i tifi t ti Al EE ( EXP) k t t Dividing by more than one number: Always use ÷ to designate a number on the bottom 1 Powers & negative numbers: To make something Combining operations: Always use parenthesis around addition and subtraction when combining with other operations 3 Scientific notation: Always use EE (or EXP) key to enter in scientific notation 2 64/67 Powers & negative numbers: To make something negative, keep the negative sign out of the exponent operation 4
  33. 33. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 33 Trigonometric functions are Trigonometric functions are Algebra is often used to solve for Algebra is often used to solve for After working hard on a problem, avoid After working hard on a problem, avoid Learning Summary used in physics. used in physics. solve for variables in physics. solve for variables in physics. common calculator mistakes! common calculator mistakes! Results ofResults of 65/67 Follow rules for working with exponents & scientific notation. Follow rules for working with exponents & scientific notation. Results of calculations must be reported with the correct # of sig figures. Results of calculations must be reported with the correct # of sig figures. Congratulations You have successfully completed the core tutorial Basic Math for Physics Rapid Learning CenterRapid Learning Center
  34. 34. AP Physics Rapid Learning Series - 02 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 34 Rapid Learning Center Wh t’ N t Chemistry :: Biology :: Physics :: Math What’s Next … Step 1: Concepts – Core Tutorial (Just Completed) Step 2: Practice – Interactive Problem Drill Step 3: Recap – Super Review Cheat Sheet 67/67 Go for it! http://www.RapidLearningCenter.com

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