1 AP Physics C: Prerequisite Topics 2018-19 Summer Assignment Dr. DePalma Consistency and validity of units in equations 1. Using the given units for each variable, determine if each equation is valid. mass (m): kg time (t): s distance (x): m velocity(v): m/s acceleration (a): m/s 2 a. x f = x i + vt b. x f x i = v it + ½ at 2 c. a = x/t 2 d. x 2 = ½ gt e. v = at f. v 2 = 2ax 2. Newton s 2 nd law states acceleration, in m/s 2 is directly proportional to the force acting on an object, and is inversely proportional to the objects mass, in kg. What are the dimensions of force in meters, kg, and seconds? 3. Velocity is related to acceleration and distance by the equation v 2 = 2ax p. Find the power p that makes this equation dimensionally consistent. Common metric prefixes: 4. Write the corresponding multiple of 10 deci (d) centi (c) milli (m) mega (M) micro (µ) nano (n) kilo (k) 5. A basketball coach insists that his players be at least 180 cm tall. Would a player of height 5 ft 11.5 inches tall qualify for the team? 6. A football field is 100 yards long. Express this distance in millimeters
2 7. Convert each of the following into the indicated units: a. 5 m to ft b. 1.2 ft/s to m/s c. 9.8 m/s 2 to ft/s 2 d. 535 kg to mg e. 5 mm = km f. 65 mi/hr = m/s g. 1 g/cm 3 = kg/m 3 8. A computer can do 2 gigacalculations per second. How many calculations can it do in a microsecond? 9. The micrometer (1 um) is often called a micron. a. How many microns make up 1 km? b. What fraction of a centimeter equals 1 um? c. How many microns are in 1 yard? 10. Earth is approximately a sphere of radius 6.47 x 10 6 m. a. What is its circumference in km? b. What is its surface area in square km? c. What is its volume in cubic kilometers? 11. The Star of Africa, a diamond in the royal scepter of the British crown jewels, has a mass of 530.2 carats. Given that 1 carat = 0.2 g, and 1 kg = 2.21 lb, what is the weight of this diamond in pounds?
3 Errors 1. Personal mistakes made by the experimenter in taking/recording data, calculations 2. Systematic results are consistently above/below true value Flaw in apparatus (incorrectly calibrated) Failure to take all important variables into account (neglecting friction) 3. Random produced by unpredictable and unknown variations. In principle, all personal and systematic errors can be eliminated, but there will always be random errors in any measurement. Mean the best approximation to the true value (the average value) Standard deviation determines precision Probability theory states that approximately 68.3% of all repeated measurements will fall within 1 standard deviation, and 95.5% should fall within 2 standard deviations from mean. Standard error precision of the mean 12. Physicist Enrico Fermi once pointed out that a standard lecture period (50 minutes) is close to 1 microcentury. a. How long is a microcentury in minutes? b. Find the percentage difference from Fermi s approximation by using the following;. percentage difference = actual approximation x 100% actual
4 Measurement Activity Two metersticks are placed along a table as shown below, and 5 readings for length and width are recorded in Data Table 1, estimated to 4 decimal places. The length for each trial is found by subtracting x1 from x2, and the width is found by subtracting y1 from y2. x2 x1 1. Find these values and record them in Data Table 2. 2. Calculate the mean length (L) and mean width (W) of the table, and record on the data sheet (the number of sig figs will be determined after step 4 below) 3. Calculate the standard deviation from the mean for length (σl) and width (σw). Record to 2 sig figs 4. Calculate the standard error for the length (αl) and width (αw) To determine how many significant figures for the mean values of length and width: The first decimal place that is significant in αl and αw defines the decimal place that is uncertain in L and W. Therefore, make that decimal place the least significant figure in the calculation of L and W. For example: suppose 10 measurements have a mean length of 1.352543 m, and αl = 0.000086 m. The first number that is significant is the fifth decimal place, therefore, L would be recorded as 1.35254 m. Go back to your values for L and W, and record to the correct number of significant figures. 5. Calculate to 2 sig figs the standard deviation from the mean of the area (σa) of the table, and the standard error (αa) using the following equations:
5 6. Using the equation A = L x W, calculate the area of the table. You will determine the number of significant figures in the result in 2 different ways: a. For A1, Determine the number of sig figs by a procedure similar to the one used above for the mean length and width. Let the first decimal place that is significant in αa be the least significant decimal place in the area. b. For A2, Use for L a value with the number of sig figs contained in each of the given values of x1 and x2, and for W, a value with the number of sig figs contained in each of the given values of y1 and y2. Take the product of those values (LxW), and use the significant figure rules for multiplication given in the notes.
6 Data Table 1 Trial x1 (m) x2 (m) y1 (m) y2 (m) 1 0.4124 3.0337 0.0473 1.2279 2 0.2243 2.8259 0.1023 1.3074 3 0.1467 2.6248 0.0835 1.3057 4 0.1574 2.7696 0.0718 1.3258 5 0.2102 2.7824 0.0861 1.2534 Data Table 2 Trial Li (x2 x1) Wi (y2 y1) 1 2 3 4 5 L = m W = m σl = m σw = m αl = m αw = m σa = m 2 αa = m 2 A1 = m 2 A2 = m 2 Answer the Question on the back of this sheet.
7 1. What percentage of the given length measurements falls within one standard deviation for the mean length? What percentage falls within 2 standard deviations? 2. What percentage of the given width measurements falls within one standard deviation for the mean width? What percentage falls within 2 standard deviations? 3. According to the theory of random errors, what percentage would be expected to be within one standard deviation? Two standard deviations? 4. Based upon your answers above, are your data reasonably consistent with the assumption that only random errors are present in the experiment? State clearly the basis for your answer 5. The two values for the area represent 2 different approaches to determination of the uncertainty in a calculated quantity. The value given as A1 represents the most precise determination of the uncertainty based upon the application of statistical theory to assumed random measurement errors. The value given as A2 assumes that the uncertainty is in the least significant digit for each individual measurement, but does not explicitly state what the uncertainty is in that digit. a. If it is possible to do so for these measurements, state whether A1 or A2 is the most precise. b. Assuming that A2 is only an approximation to the more correct A1, can you state whether A2 overestimates or underestimates the uncertainty compare to that predicted by A1? State clearly the basis for your answer.
8 Vectors 1. The Dognapping The evil members of SAP (society against physics) have dognapped Dr D s trusty 3-legged dog, Tripod, and are demanding a ransom of one million autographed pictures of Dr. D plus $5. The ransom note leads Dr. D on the following path: From home, she goes 100 km due east, turns, and then goes 500 km in a direction 30 north of east. Next, she turns 45 south of east and moves 100 km, then travels 350 km due north. A bucket of half eaten KFC is found, and Dr D knows she is on the right path since this is Tripods favorite food! After finishing off the chicken, she travels 300 km in a direction of 40 north of west, then 200 km at 30 south of west. Finally, after dropping the ransom in a container marked SAP random-drop Here, Dr D finds Tripod in perfect shape, but tired and resting, propped against a wall. How far away, and in what direction, are Dr D and Tripod from home? 2. War Games Dr D is participating in maneuvers with the local reserve, and even though they are using rubber bullets, she is not convinced that she would rebound from a direct hit (Dr D is allergic to bullets, she breaks out in holes). Deserted by her trusty dog Tripod (he had to turn in his dogtags when it was determined that he couldn t salute without falling over), Dr. D is all alone, abandoned by her trusty AP Physics students. Sure, she had bombed in a school play, and often shot off her mouth, but firing guns at tanks was a completely new experience! Soon she spots a tank 200 m away moving at 15 m/s on a line perpendicular to her line of sight. She knows that the speed of the bullet is 300 m/s. At what horizontal angle with her line of sight must Dr. D aim in order to hit the tank?
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10 Intro to Calculus Directions: Visit http://www.zweigmedia.com/tuts/index.html?lang=en, and click on the tab Online Tutorials if you need help with the following problems. Derivatives 1. 2. 3. 4. 5.
11 Visualizing Derivatives Use the graph of v(t) shown below to answer questions #6-11. 6) Find v(t) dt for 0 to 3. Show work below, indicate the appropriate units and include an illustration of this quantity on the graph above. 7) Find the value of v(t) dt for t = 0 through t = 10 s 8) If F(t) is a function such that F(0) = 0 and F (t) = v(t), find the intervals where F(t) is increasing: concave up: decreasing: concave down: parabolic: linear: 9) Use the information in questions #2 and 3 to sketch an accurate graph of F(t). Use graph paper, Label the axes and indicate the units. 10) How would the above graph of F(t) change if F(0) = 2 instead? 11) What does F(t) represent in the real world?
12 Integration 1. 2. 3. 4. 5.