Name: Email address (write legibly): AP Physics 1 Summer Assignment Packet 2 Read through the Symbols, Units, Equations, and Sample Problem given in the instructions packet. The worksheets included in this packet are to be submitted by July 24 th. They are: Scientific Notation Significant Figures Trigonometry Review Falling Masses Activity (to be performed after completing Topic 1 from The Physics Classroom) Read through and complete the work in Appendix A - Mathematical Review. (Do not hand in your work from this.) Other materials due by July 24 th : Topics 1 and 2 from The Physics Classroom Email to Mr. Sadowsky Read the instructions packet for all the details about these assignments.
Name: Email address (write legibly): AP Physics 1 - Scientific Notation Scientific Notation: When a number is in scientific notation, the form is of a number multiplied by 10 raised to some power or: A x 10 B (where A can be a decimal and B is an integer) Expanded Notation: When a number is in expanded notation, it is not multiplied by a factor of 10 anymore. OR, you can consider it to be multiplied by 10 0, which will not change its value because 10 0 is mathematically equivalent to the number 1. (Ex.: 3.14 = 3.14 x 10 0 0.042 = 0.042 x 10 0 ) Since it is often advantageous to put numbers in scientific notation, here are the rules: Rules: I) Only one nonzero digit (1-9) before the decimal point. II) When you move the decimal point to the Left, the exponent gets Larger. III) When you move the decimal point to the Right, the exponent gets Reduced. So, when you put a number in scientific notation, slap a 10 0 on the end, and then move the decimal point until it conforms to Rule 1. When you expand a number out of scientific notation, move the decimal until the exponent becomes zero, and then get rid of the 10 0. (Simply don t write it.) (Answers are given on the back of the next page.) (Are you boxing in your answers, like you should?) Put the following in scientific notation: 1) 0.0025 Expand the following: 3) 2.71828 x 10 0 2) 6 4) 5.67 x 10-21 IV) When adding or subtracting numbers in scientific notation, the factor of ten (exponent) must be the same before you add or subtract. This may mean that you, temporarily, need to change a number so it is not in scientific notation anymore. 5) The radius of the Earth is 6,378,000 meters. Felix Baumgartner, an Austrian skydiver and member of the Red Bull Stratos project, dove from 127,852 feet above the Earth s surface (that s about 39,000 meters). At the top of the skydive, how far was Mr. Baumgartner from the center of the Earth? Put your answer in scientific notation.
V) When multiplying numbers in scientific notation, multiply the numbers in front and add the exponents. (think: 100 x 1000 = (1 x 10 2 ) x (1 x 10 3 )= 1 x 10 5 = 100,000) When dividing, divide the numbers and subtract the exponents. (think: 100 / 1000 = (1 x 10 2 ) / (1 x 10 3 ) = 1 x 10-1 = 0.1) (If an exponent is a negative number, keep it as a negative number.) Then, refer back to Rules 1 3 to make sure your answer is in scientific notation. (If you haven t learned the terms mentioned below yet, come back to these after doing the appropriate section of the Physics Classroom web site.) 6) The acceleration of an electron around a hydrogen nucleus is about 89,900,000,000,000,000,000,000 m/s 2. The mass of an electron is about 0.000000000000000000000000000000911 kg. Find the net force on this electron. (Hey have you been following the correct format for solving problems? In other words, are you showing all your work?) 7) The Moon (mass = 73,500,000,000,000,000,000,000 kg) circles around the Earth once every 27.322 days (2,360,600 seconds). Its distance from the Earth (radius of rotation) is about 384,400,000 meters. a) Find the Moon s tangential speed (circumference / time). Put your answer in scientific notation. b) Find the Moon s kinetic energy. Put your answer in scientific notation. (There are still more problems on the back of the next page.)
Scientific Calculators and Scientific Notation A Quick Tour While it is important to know the fundamentals of scientific notation, it is also important to know the shortcuts and how to use the tools that are available to you - namely, your scientific calculator. All scientific calculators have a button on them that allows you to enter a number in scientific notation. The buttons are labeled differently on different brands of calculators. The most common labels are: EE or EXP. Find yours. If you can t locate it, ask another student for help. My calculator has the EE label, so that is what I will use in this guide when referring to that button. When you press the EE button, it takes the place of the words times 10 to the... in the name of the number you are entering. So, to enter the number 3 x 10 8 in your calculator (which is said, three times ten to the eighth ), you would press the following buttons, in this sequence: 3 EE 8 Notice that you never have to hit the digits 1 and 0! You don t have to hit the x button! You don t have to find the exponent button! It s all taken care of with the EE button. Boy, whoever decided to include this button on the scientific calculator should get a medal. When you look at the display, you might see something that looks like this: 3 8. Your calculator is NOT reading three to the eighth power. That would be the number 6561 because it would be 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3. Most calculators display numbers in scientific notation by putting the exponent up in the corner. This is due to the space restrictions of the screen. Some newer calculators may not do this because they have bigger screens. Then there s the graphing calculators. They often display numbers in scientific notation like so: 3E8. That also means 3 x 10 8. Get used to it. You need to be able to recognize how your calculator displays scientific notation. No matter how yours does it, though, here s something to remember: YOU MAY NOT WRITE A NUMBER AS 3E8 OR 3 8 WHEN YOU MEAN 3 X 10 8. I will not accept 3E8 on your tests or quizzes because that is calculator-speak. I certainly will not accept 3 8 because that is the WRONG NUMBER (remember it s 6561, as opposed to 300,000,000).
Now, problems become much easier and quicker. Try this example: Problem: The speed of light is a constant 3.00 x 10 8 m/s. How far can light travel in 1 year (about 3.15 x 10 7 s)? Since this is a distance problem, we ll use d = vt. To multiply these numbers, enter the following sequence: 3 EE 8 x 3. 1 5 EE 7 = Your answer should come out to 9.45 x 10 15 m. You can check it if you like, using the rules for scientific notation that you learned on the first sheet. Hey your calculator even puts the answer in scientific notation for you (usually)! How nice. Go back to the previous sheet and try the calculations there using the EE button. Check to make sure you are familiar with it and its uses. For example, to enter numbers with negative exponents just takes a few more keystrokes, in a logical order. Let s take the mass of the electron, which is 9.11 x 10-31 kg: 9. 1 1 EE +/- 3 1 It also works if you hit the +/- button after entering the exponent on my calculator. Play around with yours. What DOESN T work is hitting the +/- button any time before hitting the EE button. Try this and you ll see that you don t get 9.11 x 10-31 ; you get 9.11 x 10 31. Make sure that you are careful and that you put the negative sign where you want it. Do the problems below (answers follow). If you haven t learned the terms mentioned yet, come back to these after doing the appropriate section of the Physics Classroom web site. 8) Find the time it takes for an electron to travel a distance of 3.10 x 10-3 m when it starts from rest and experiences an acceleration of 7.68 x 10 10 m/s 2. Put your answer in scientific notation. 9) An ion experiences a net force of 6.00 x 10-14 N. It starts from rest and acquires a speed of 3.40 x 10 5 m/s in a time of 3.76 x 10-8 seconds. Find its mass. Put your answer in scientific notation. Answers 1) 2.5 x 10-3 2) 6 x 10 0 3) 2.71828 4) 0.00000000000000000000567 5) 6.417 x 10 6 m 6) 8.19 x 10-8 N 7) a) 1.023 x 10 3 m/s b) 3.85 x 10 28 J 8) 2.84 x 10-7 s 9) 6.64 x 10-27 kg
Name: AP Physics 1 Significant Figures Accuracy describes how close a measured value is to the true value of the quantity measured. Problems with accuracy are due to error, whether there is a problem with the measuring device (equipment error) or with the person making the measurement (human error). Precision refers to the degree of exactness with which a measurement is made and stated. Precision describes the limitations of the measuring instrument. Significant Figures (or sig figs ) those digits in a measurement that are known with certainty plus the first digit that is uncertain. Generally, a measurement, including the appropriate unit, is taken to onetenth the smallest division of the instrument being used (that is the estimated digit). Significant figures help keep track of imprecision. Rules for counting significant figures: 1. Nonzero digits are significant. 2. Zeros between other nonzero digits are significant. 3. Zeros in front of nonzero digits are not significant. 4. Zeros at the end of a number and also to the right of the decimal are significant. 5. Zeros at the end of a number but to the left of a decimal are significant if they have been measured or are the first estimated digit; otherwise, they are not significant. Generally, they are treated as not significant. 6. Use proper scientific notation when possible all digits are significant. Examples: Value Number of Sig Figs Rule(s) Value Number of Sig Figs Rule(s) 1 1 1 10 1 5 1.0 2 4 100 1 5 1.00 3 4 101 3 2 0.1 1 3 1010 3 2,5 0.10 2 3,4 1010.1 5 2 0.100 3 3,4 1010.0 5 2,4 0.01 1 3 1.0 x 10 7 2 6 Rules for calculating with significant figures: Addition/Subtraction The final answer should have the same number of digits to the right of the decimal as the measurement with the smallest number of digits to the right of the decimal. Multiplication/Division The final answer has the same number of significant figures as the measurement having the smallest number of significant figures. You may assume that numbers not measured (e.g., constants and conversion factors) have an infinite number of significant figures. After performing the calculations and determining the proper number of sig figs for a particular problem, your answer should also be rounded correctly. (5 and above rounds up; 4 and below rounds down) But, ONLY ROUND ONCE. For example, if you round your answer to 3 sig figs, and then realize you really needed only 2, go back to the original calculation and see how it should be rounded. (In other words, you can t go from an answer of 8.445 8.45 8.5. To 2 sig figs, 8.445 rounds to 8.4. An answer of 8.5 would now be incorrect.) (continued on the back)
(continued from the front) Problems: (Answers don t peek yet! are at the bottom of the page.) 1) Write in the number of significant figures for each measurement and list which rule(s) tells you why the zeroes (if there are any) are significant or not. Measurement Number of Sig Figs Rule Measurement Number of Sig Figs Rule a) 3.0025 s e) 2.0 x 10 2 kg b) 0.0008 s f) 300,000,000 m/s c) 0.000800 s g) 3 x 10 8 m/s d) 57.00 g h) 8,990,000,000 N. m 2 /C 2 2) Your calculator reads 57.08506224. Assuming this is a distance measured in meters, round this value to: a) 3 sig figs: b) 2 sig figs: c) 4 sig figs: d) 1 sig fig: 3) Your calculator reads 0.003737373. Assuming this is time measured in seconds, round this value to: a) 3 sig figs: b) 2 sig figs: c) 4 sig figs: d) 1 sig fig: 4) In a problem, you need to divide 2.2 m by another number. Perform the calculation and round your answer to the correct number of sig figs if that second number is: a) 5 s: b) 5.15 s: c) 5.2 s: d) 5.146 s: 5) Albert walks 5.0 meters in 3.62 seconds. Find his average speed. 6) Albert walks 10.00 meters in 3 seconds. Find his average speed. Answers 1) a) 5 b) 1 c) 3 d) 4 e) 2 f) 1 g) 1 h) 3 2) a) 57.1 m b) 57 m c) 57.09 m d) 60 m 3) a) 0.00374 s b) 0.0037 s c) 0.003737 s d) 0.004 s 4 ) a) 0.4 m/s b) 0.43 m/s c) 0.42 m/s d) 0.43 m/s 5) 1.4 m/s (not 1.38 m/s, not 1.38121547 m/s) 6) 3 m/s (not 3.3 m/s, not 3.33 m/s, and NEVER 3.3 m/s) (repeating decimals imply infinite precision, which CANNOT HAPPEN!)
Why do we deal with significant figures? It all comes down to: how precise is your measurement? In math, you deal with abstract numbers that don t necessarily have meaning. Therefore, the answers you get by manipulating the numbers can be infinitely precise. 42 = 42.0 = 42.00= 42.00000000000 and so on. All those numbers have the same value. But in science, we deal with quantities that represent real measurements. In the real world, there is no such thing as an infinitely precise number measuring devices simply have a limit to their precision. If you use a stopwatch to measure an elapsed time, and the stopwatch reads to the hundredth of a second, you might measure a time as 42.00 seconds. Your measurement has 4 sig figs. You absolutely cannot say that is the same thing as 42.000 seconds because the stopwatch does not read to the 3 rd decimal place, and so you have no idea what the next digit is. It could have been 41.998 or perhaps 42.003. You simply can t tell, so your measurement is limited to 4 sig figs what the stopwatch reads, in this case. If you measured a shorter time as 3.24 seconds, you now have 3 sig figs in that measurement. Now, here is the important part: any calculation done with your measurement is limited by the precision of the measurement. As an example, let s say you measure the distance an object moved as 81 meters and the time it took to do so as 42.00 seconds. You might have measured the distance to only 2 sig figs for a variety of reasons: the measuring device only reliably measures to the nearest meter, the event happened too quickly for you to be confident in how far the object moved, the object itself has a significant size to it (so it would not be of any use to measure more precisely than the size of the object), etc. If you want to find this object s average speed, you would get 1.9 m/s. Your answer has 2 sig figs because, even though the time has 4 sig figs, the distance only has 2, and that limits your answer to 2, as well. Now, let s see what range of values you could have gotten. The 81 m measurement is an estimation to the nearest meter, so the true distance the object moved may be anywhere from 80.5 to 81.4 (both would round to 81), making your measurement 81 m +/- 0.5 m. The time could be anywhere from 41.095 to 42.004, making your measurement 42.00 s +/- 0.005 s. Let s take the worst case scenarios: Pairing the smallest possible distance of 80.5 m with the largest possible time of 42.004 s, your answer would be 1.92 m/s. If the distance was really 81.4 m (the largest) and the time was 41.095 s (the smallest), your answer would be 1.98 m/s. If you had originally reported the average speed to 3 sig figs, you would have gotten 1.93 m/s. You can see that all of these agree on the ones digit and the tenths digit, but there is uncertainty in the hundreds digit. With this uncertainty, that 3 rd significant figure has no meaning. Your calculation is only good to 2 sig figs because, to 2 sig figs, there is agreement from the smallest possible answer to the largest possible answer. Now, in reality, if you had to estimate the distance to the nearest meter, you may not even be certain about that. Maybe the distance was really 80 m or 82 m. Again, let s take the worst case scenarios. With 80 m and 42.004 s, you get 1.9 m/s. With 82 m and 41.095 s, you get 2.0 m/s. You can t even really be certain about the 9 in the tenths place. It clearly makes no sense to report any decimal places past that. This is why adhering to the rules of significant figures matters. Your answer cannot be more precise than your least precise measurement. To go beyond that level of precision is to report answers that have no value there will always be estimations and uncertainty.
Name: Trigonometry Review The trigonometric functions (sine, cosine, and tangent) are ratios of sides of a right triangle. The diagram below shows a generic right triangle, with its sides labeled: opposite (O), adjacent (A), and hypotenuse (H). The right angle is designated by the square in the corner. One of the other angles is labeled by the symbol for angle the Greek letter theta ( ). Trigonometric Functions sine: sin = O H cosine: cos = A H H O tangent: tan = O A A opposite (O) the side of the triangle across from the angle ( ) in question. adjacent (A) the side of the triangle next to the angle ( ) in question. hypotenuse (H) the side of the triangle across from the right angle; necessarily, the longest side of the triangle. The easy way to remember the trig. functions is this: SOHCAHTOA. S ine O opposite H ypotenuse C osine A djacent H ypotenuse T angent O opposite A djacent Another helpful equation when dealing with right triangles is the Pythagorean Theorem, which looks like this: a 2 + b 2 = c 2 where a, b, and c are the three sides of a right triangle, with c being the hypotenuse. If you want to use the symbols previously defined, it would look like this: O 2 + A 2 = H 2
The following problems will give you practice using the trig. functions and using your calculator. Before you start, you need to make sure your calculator is in the correct mode. There are various ways to measure angles; for this class, we will use degrees ( o ) almost exclusively. Make sure your calculator is in degree mode. If you need help doing this, consult the documentation for your calculator. Answers for these problems are on the back of the next page. For the problems 1-4, you will have to pick the correct trig. function to use and then rearrange it to solve for the requested side of the triangle. The angle,, stays with its function. In other words, if it is written sin, that is NOT sine times. To start you off, if you were to rearrange the cosine function to solve for the adjacent side, it would look like this: A = Hcos. 1) Find the horizontal side (x): 2) Find the vertical side (y): 10.0 m y 2.00 N x 30.0 o 47.0 o 3) Find the hypotenuse (H): 4) Find the horizontal side (x): H 22 o 12.0 m 5.0 m/s x 15.0 o 5) A 5.5 N force acts at 22.0 o below the horizontal. Find the vertical and horizontal components of the force. (Form a right triangle with the given force as the hypotenuse. The legs of the triangle are the components.) 22.0 o 5.5 N 6) A soccer ball is kicked at 14.0 m/s at 59 o above the horizontal. Find the vertical and horizontal components of the velocity. 14.0 m/s 59 o
The inverse trig. functions are used to find the angle measure ( ) if you already have two sides of the triangle. They look like this: Inverse sine Inverse cosine Inverse tangent = sin -1 O H = cos -1 A H = tan -1 O A To use these on your calculator, you probably have to press a shift or 2 nd function key and then the trig. function key. Ask for help if you cannot figure out how to get the inverse trig. functions. For most calculators with a single-line display, you first have to determine the ratio of sides (do the division first) and then evaluate the inverse trig. function. For most multi-line display calculators, you type it in as you see it. Again, remember to have your calculator in degree mode. For each triangle below, find the angle,. You need to decide which trig. function to use, depending on which sides of the triangle you have. 7) 8) 6.04 m 10.0 m 5 m 6.18 m 9) 3.0 m/s 5.0 m/s 10) 14 m 18.0 m 11) 9.00 m 12) 10.00 m 5.10 m 8.14 m
Answers 1) 8.66 m 2) 1.36 N 3) 13 m (2 sig figs, right?) 4) 19 m/s 5) vertical: 2.1 N horizontal: 5.1 N 6) vertical: 12 m/s horizontal: 7.2 m/s 7) 44.3 o 8) 30 o (how many sig figs?) 9) 53 o 10) 51 o 11) 42.0 o 12) 38.8 o
AP Physics 1 Summer Assignment Falling Masses Activity Hypothesis: You will be plotting a graph of distance fallen vs. time for a falling object. Thinking about what the slope of such a graph represents and what happens to that quantity as the object falls (and thinking about the reading you have been doing on physicsclassroom.com), sketch the shape of the graph you expect to get: d Materials: a dollar bill length measuring device(s) (calibrated straight edge, meter stick, tape measure, etc., as long as it measures in metric units or you are willing to convert) towel (optional) stopwatch (the one on your cell phone is just fine) 3 spheres of various sizes (marbles, bouncy balls, sports balls, etc.); try to use two of the same size but different masses t Procedure: 1) Measure the length (longer dimension) of a dollar bill in centimeters. Record it on the Data/Results page provided. 2) You need a partner for this step. Hold the dollar bill the long way vertical so that it hangs between your partner s thumb and forefinger with his/her fingers positioned at the halfway mark. Without telling your partner when you will do it, drop the dollar bill. See if your partner can close his/her fingers in time to catch the bill. (Most people can t react quickly enough.) 3) Using kinematics equations, calculate the time it takes for half the dollar bill s length to pass between your partner s fingers. This is an approximation of human reaction time. (what units should the distance be in?) SAFETY: For the next part, you will be dropping objects from various heights. Make certain everyone around you knows what is going on so no one steps into the path of a falling object and gets hurt. Inform your parents about this experiment so they can make suggestions that will keep you (and their floors) safe. 4) Drop the smallest of your spheres (marble, bouncy ball, etc.) from rest from each of the given heights and measure the time to fall. Get a partner to help you, especially with the larger heights. Put a folded towel down where the sphere will hit so the floor does not get damaged and, if using a fragile object (such as a glass marble), the sphere will not get break. Alternatively, perform the experiment outside in a grassy area. Use a stopwatch to time how long it takes to hit the ground. This will be Trial 1. Do it again. This time will be Trial 2. Average the two trials. 5) Repeat step 4 with a) another sphere of approximately the same size but different mass b) a large sports ball, such as a soccer ball or a basketball (not a bowling ball you ll hurt yourself or others). 6) For each object, complete the remaining columns on the data chart. There is no need to show work for those calculations.
Data/Results: Length of dollar bill: Calculation for human reaction time (show work here): Object: Distance Fallen (m) Time to Fall (s) Trial 1 Trial 2 Average 2 (Distance Fallen) (m) (Average Time) 2 (s 2 ) 2.00 1.80 1.60 1.40 1.20 0.80 0.40 0.20 0
Object: Distance Fallen (m) Time to Fall (s) Trial 1 Trial 2 Average 2 (Distance Fallen) (m) (Average Time) 2 (s 2 ) 2.00 1.80 1.60 1.40 1.20 0.80 0.40 0.20 0 Object: Distance Fallen (m) Time to Fall (s) Trial 1 Trial 2 Average 2 (Distance Fallen) (m) (Average Time) 2 (s 2 ) 2.00 1.80 1.60 1.40 1.20 0.80 0.40 0.20 0
Graphs: (graph paper is provided in this packet) Before plotting any graphs, carefully read the Expectations for Graphing part of the Summer Assignment. You will plot 4 separate graphs, as listed below, on 4 separate pieces of graph paper. Make sure the title of each graph indicates the object used for that data set. Plot Distance Fallen vs. Average Time for the first object only. Plot 2 (Distance Fallen) vs. (Average Time) 2 for the first object. Plot 2 (Distance Fallen) vs. (Average Time) 2 for the second object. Plot 2 (Distance Fallen) vs. (Average Time) 2 for the third object. Analysis: (answer on these pages) 1) What shape was the distance vs. time graph for the marble? Does it match your hypothesized graph? If not, YOU MAY NOT CHANGE YOUR HYPOTHESIS. Instead, explain what you have learned. 2) What does the slope of a distance vs. time graph represent? 3) So, why is it natural to expect this particular shape for distance vs. time for this situation? (What was happening to the marble s motion as it fell?) 4) The equation of motion that applies to uniform acceleration is: d = v o t + ½ at 2 Since you dropped all the objects from rest, v o = 0 m/s. Crossing out the v o t term and multiplying both sides by 2 yields: 2d = at 2. Notice that the remaining graphs you plotted had 2d on the vertical axis and t 2 on the horizontal axis and, if all went well, you should have gotten very good straight-line trends when you plotted these three graphs. (If you didn t, then maybe it s time to go back and do the experiment again....) The general algebraic equation for a straight line is y = mx + b. Fitting the rearranged kinematics equation to this form shows: 2d = at 2 it might help if it is written so: (2d) = (a)(t 2 ) y = mx + b y = m x + b So, what does the slope of your 2d vs. t 2 graph represent? 5) Assuming the dropped objects were in free-fall, what is the expected value for the slope of the graph?
6) If you have not already done so, calculate the slope of each 2d vs. t 2 graph. Show your work on the graph. (If your pattern is not a good straight line, find the best straight line that goes through the pattern of plotted points and use that.) (Or again, maybe you should re-do the experiment....) Object Slope 7) What challenges did you have to overcome in performing this experiment? What difficulties did you encounter, based on the size of the object? 8) When measuring the starting height of each object, what part of the object did you measure to (center, top, bottom, random)? Upon reflection, was that the best choice? If so, what mistake might another student have made? If not, how would you change it to better correspond to the distance the object fell? 9) When trying to establish a pattern, how many data points do you think are necessary? 3? 5? More? Why are you told to take data for 8 different heights in this experiment?
10) Which is easier measuring long times or short times? How does human reaction time play a role in this? Which data points do you have the most confidence in? 11) Why are multiple trials done for the same height? 12) For each object, calculate the percent discrepancy, using your answer to Analysis number 5 as the accepted value and your slope as the observed. Show your work in the space below. % discrepancy = observed accepted x 100 accepted 13) Evaluate your results. Do you feel, within experimental error, this experiment showed objects in freefall? For some, not others? What factors were present that you could not control and how did they affect each object?