Hanover Park Regional High School District

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1 Hanover Park Regional High School District Dear A.P. Physics 1 Student and Parents, June 2017 Welcome to A.P. Physics 1! In the year ahead, we will cover many exciting topics in physics in preparation for the Advanced Placement Physics 1 exam. A.P. Physics is both an extensive and challenging course, and in keeping with this a summer assignment has been implemented to move more quickly through the curriculum. The A.P. Physics 1 course is designed for the advanced student to learn about several areas in physics, as well as to improve their general science and reasoning skills. This course is comparable to an introductory algebra-based college physics course. Colleges give up to 15 hours of credit for grades of 3, 4, and 5 on the final A.P. exam to be taken in May. Each student will receive a copy of the summer assignment. Students will be asked to read and answer questions in the attached packet. The material in this packet should be a review of skills learned in Algebra II/ Trigonometry and previous science courses. Many questions use these skills and ask you to apply them in a physics context as you will need to do throughout the year. If you find the math difficult, you may find additional online resources such as Khan Academy to be helpful. ALL WORK MUST BE NEATLY SHOWN, AND ALL ANSWERS SHOULD BE BOXED/CIRCLED! Work may be done on the packet itself or attached on separate, neatly organized notebook paper. The assignment must be completed and brought to the first day of class (Thursday September 7 th ). Students have the entire summer to complete the assignment so INCOMPLETE OR LATE ASSIGNMENTS WILL NOT BE ACCEPTED (no excuses)! During the first full week of school, students will be tested on the material covered in this packet. Time will be taken during the first few days of school to review and address any questions that arise in the packet. It is critical that the students are comfortable with this material in order to succeed in the course as the skills in this packet are used throughout the year. I look forward to working you in the forthcoming school year. Please feel free to contact me if you have any questions. I will get back to you as soon as possible. Sincerely, Thomas O Dell Hanover Park A.P. Physics Teacher todell@hpreg.org Michael Kleeman Whippany Park A.P. Physics Teacher mkleeman@hpreg.org

2 2 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Welcome to AP Physics 1! This packet is designed to review skills and concepts that you should be familiar with from previous science and mathematics courses. Section 1: When are we ever going to actually use this? You ve probably asked this question of yourself at least a few times in the past. In physics, the question should be easy to answer. Physics is all about observing the way the world works and trying to make sense of it. Once we find an explanation for how things work, we can move on to why things work and then even into engineering to make them work better. Physicists take the tools you learn in math class and apply them to every day scenarios, and this means that to succeed in physics, you need to be familiar with a few key concepts in math. Perhaps the most important tool physicists take from math is the proportion. Physicists love to find relationships between two different things as a way to make sense of them. For example, if you step on the gas pedal in your car for one second, you know that you ll go faster. But if you step on the gas pedal for two seconds instead (doubling the time), do you double the increase in speed? Quadruple it? Or something else entirely? That s the type of question that a physicist tries to answer to learn more about how things work. In the table below are four common relationships that you ll see between two variables. In the table they re y and x, but they could be anything. In our example above they would be change in speed and time. (The symbol means proportional to, and that means that the two sides of the proportion change together. In y x 2, y would double when x 2 doubles, y would triple when x 2 triples, etc. ) Linear Quadratic Inverse Square-Root y x y x 2 y 1 x y x Let s say you re on the highway with the cruise control on at 60mph. You want to find the relationship between the amount of time that you drive and the distance you cover. If you travel for one hour, you ll go 60 miles, and if you travel for two hours, you ll go 120 miles. You could extend this to three hours and 180 miles and so on like in the data table to the right. If you compare the first and second data points, you ll notice that you doubled the time and you got double the distance. If you compare the first and third data points, Time (hrs) Distance (mi) you tripled the time and tripled the distance. The pattern here is that whatever change you made to the first column (doubled, tripled, etc), you make the same change to get the second data point. Any time you see this pattern, you have a linear relationship. At this point we can say that distance (d) is directly proportional to time (t), or in symbols d t.

3 Distance (m) Distance (miles) AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 3 If you graphed the data, you would get a straight line like the first graph in the table above. As soon as we Distance vs. Time see a straight line, we can identify the proportion, or 400 relationship between the two variables, as linear. We can now put a best-fit line on the graph and find its 300 d = 60t equation, which is d=60t. If you compare the 200 equation we just found to the proportion, they look 100 similar. The variables are in the same place, and we 0 just put the 60 in to make it an equation. The 60 is a constant, or multiplier, that makes the equation Time (hours) work out in exactly the right way. The proportion is a general guide to how distance and time are related and will be your first step, but the equation tells you exactly how to go from knowing a time to knowing a distance, and will give you an actual numerical value. Proportion d t Equation d=60t Now let s say you want to know the relationship between the amount of time you step on the gas pedal in your car and the resulting distance that you cover. This is different from the last example because now you re accelerating as you travel instead of just traveling at a constant speed. You take some data and get the table to the right. This doesn t fit the same pattern as the last example. If we compare the second and third line, we double time and get four times the distance, and for the second and fourth line we triple time and have nine times the distance! All we know right now is that we don t Time (seconds) Distance covered (m) have a nice linear relationship anymore. To find out what we do have, we ll need to graph it. Since we know it s not linear, we shouldn t expect a straight line Distance vs. Time Time (sec) Comparing to the table above, this looks like a quadratic relationship. You can draw a smooth curve through the data points to make it look like half of a parabola. This means that in this case, distance is proportional to time squared, or d t 2. We can see this by comparing data points (2,12) and (4, 48). If we square our time values, we have (4,12) and (16, 48) as our ordered pairs that represent (t 2, d).. Now t 2 and d both went up by a factor of 4, which means t 2 and d are changing together, the definition of a proportional relationship. Getting the equation for this graph would be the next step, but for graphs that are not linear this is a topic we ll get to in the next section. The equation for this one happens to be d = 3t 2.

4 4 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Graphing & Best-Fit Lines: Step 1: Find Slope Pick two points that are exactly on the line. Data points that are exactly on the line, like (2,6) and (22,16), will work, but you can also pick non-data points like (6,8) and (18,14). We ll use these last two points. slope = y 2 y 1 x 2 x 1 = = 6 12 = 1 2 You should be able to draw and find the equation of the line to the left. The asterisks are data points that may have been collected in an experiment, and the straight line is the line of best fit, or the straight line that gets as close as possible to as many data points as possible. To find the actual equation, we need two pieces of information: the slope and the y-intercept. We want our equation in standard y=mx+b form. Step 2: Find Equation Use point-slope formula y y 1 = m(x x 1 ) Pick a point on the line like (6,8) y 8 = 1 (x 6) 2 y 8 = 1 2 x 3 y = 1 2 x + 5 You can also plug a point into y=mx+b and solve for b Non-Linear Graphs: You can still find the equation of a graph that s not linear, it just requires a few extra steps. These steps are called linearization because they take your non-linear graph, make a change so it becomes linear, and then let you apply the same steps as above to find the equation. Steps for linearization: 1.) Recognize the non-linear relationship on the y vs. x graph (quadratic, inverse, etc.) 2.) Re-graph a proportion that is linear (y vs. x 2, y vs. 1 x, etc.) 3.) Draw a best fit line and find the equation. Let s use our example of a non-linear graph from page 2. The graph of distance on the y axis and time on the x-axis (a d vs. t graph) is quadratic, and we don t have the tools to find the equation for that graph yet. Once we recognize the relationship shown on the graph as quadratic, it tells us that d is directly proportional to t 2. If we make a graph of d vs t 2, it should be linear because graphing any two quantities that are proportional always makes a linear graph. Check out the work on the next page to see exactly how to get to the equation:

5 Distance (m) Distance (m) AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 5 Time Distance (seconds) covered (m) Distance vs. Time Time (sec) This graph of d vs t is quadratic. Time 2 Distance (seconds) covered (m) Distance vs. Time^ Time^2 (s^2) This graph of d vs t 2 is linear, and we can now find its equation. The equation has the general form of y=3x. Checking our axes, the y-axis represents d and the x-axis represents t 2. This turns y=3x into d=3t 2. The general rule for linearization is that you will always want to change your x-axis values to match the relationship. If the graph is quadratic, put x 2 on the x-axis. If the graph is inverse, put 1 on the x-axis. x If the graph is a square root graph, put x on the x-axis. When you re-graph, you will get a linear relationship that shows a direct proportion. See another example on the next page.

6 Speed (m/s) Speed (m/s) 6 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Given below is data for the maximum speed a car can safely travel around a curve measured at different radii. Your job would be to find the exact mathematical relationship between the data. Radius (m) Speed (m/s) Speed vs Radius (V vs. R plot) Radius (m) The graph above shows a square-root function. It tells us that v r. This means that v is proportional to r, and the next step would be to make a graph with v on the y-axis again, but r on the x-axis instead. r^ (1/2) v 0 0 v vs. r 1/ Square root of radius This graph is linear, and has an equation of the general form y=4x. Checking our axes, we see the y-axis represents v (speed) and that the x-axis represents r. This changes y=4x into v = 4 r

7 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 7 Graphical Methods-Summary A graph is one of the most effective representations of the relationship between two variables. The independent variable (one controlled by the experimenter) is usually placed on the x-axis. The dependent variable (one that responds to changes in the independent variable) is usually placed on the y-axis. It is important for you to be able interpret a graphical relationship and express it in a written statement and by means of an algebraic expression. Graph shape Written relationship Modification required to linearize graph Algebraic representation As x increases, y remains the same. There is no relationship between the variables. None y b, or y is constant As x increases, y increases proportionally. Y is directly proportional to x. None y mx b As x increases, y decreases. Y is inversely proportional to x. Graph y vs 1 x, or y vs x -1 y m 1 x b Y is proportional to the square of x. Graph y vs x 2 y mx 2 b Y is proportional to the square root of x. Graph y vs. x y = m x + b Or The square of y is proportional to x. Or y 2 vs x y 2 = mx + b When you state the relationship, state how y depends on x ( e.g., as x increases, y ).

8 8 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Practical Graphing for Labs: As you learned earlier, a graph is a powerful tool for finding and demonstrating relationships between data points. It helps to be able to identify how an equation would look on a graph. For example, take the equation w = k + nt 2 Assume that you have this equation from a lab where you tested the relationship between w and t, your two variables, and that k and n are known to be constants. If you can make your equation look like y = mx + b (w = nt 2 + k in this case), it will help you identify the role of each variable on the graph. To get a linear graph, you would put w on your y-axis, t 2 on your x-axis, and that would leave you with n as your slope and k as your y-intercept. Beyond graphs: Proportions and relationships are useful tools for more than just graphs. They give us a conceptual way to talk about how two variables are related, even if we re just dealing with an equation. Using the proportion from the last graphing example (v r), we can tell that we would need four times the radius in order to double the speed. (You have to take the square root of the factor by which you change r to do the equivalent to v.) If we wanted to triple the speed, we would need to have nine times the radius. It is often useful for us to re-arrange an equation in order to figure out how a variable would change. For example, an equation for electric field is E = kq r2, where k is a constant, and E, Q, and r are variables. The equation is your rule for how each variable is related, and proportion-wise (without constants) would tell you E Q r2. This means, in words, that E is proportional to Q and E is inversely proportional to r 2. If we doubled Q and tripled r, we would change E by a factor of 2. We know this 9 because a direct proportion between Q and E tells us that if Q doubles, E will double as well (the 2 in the numerator), and an inverse square proportion tells us that if r triples, E will change by a factor of 1 3 2, or 1. Combined, the 2 and 1 make But lets say we want to know how r changes if we change E and q. The easiest way to do this is to rearrange the equation so that it becomes r=. This would look like r = kq, and proportion-wise, r E would change with Q and r would change with 1. Doubling Q would change r by a factor of 2 if E stayed the same, and doubling E would change r by a factor of 1 if Q stayed the same. E 2

9 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 9 Section 2: Algebra Skills Beyond solving a simple algebraic equation, you are also expected to be able to solve quadratic equations and two-variable systems of equations. An example of each is worked below: Memorize this! x = b ± b2 4ac 2a Quadratic Example: Solve for x 17 = 23x 4x 2 Rearrange into standard form 0 = 4x 2 23x + 17 (0=Ax 2 + Bx + C) Plug into quadratic formula x = 23 + ( 23)2 4(4)(17) = (2 4) Repeat for other root (change + to before discriminant) x = 23 ( 23)2 4(4)(17) = 4.88 (2 4) Final Answer: x= 0.871, 4.88 System of Equations You should be familiar with a few methods of solving systems from math class. The most versatile method is solving by elimination, which is shown below: 3y = 2x + 12 Example: Solve for x and y if { 2x = 5y + 8 Step-by-Step Solution: 1. Pick one of the equations to use to isolate a variable. We ll use the second in this example 2. Isolate a variable. Here, 2x is already by itself so all we need to do to get x alone is divide both sides by Plug the expression for x into the other equation. 2x = 5y + 8 2x 2 = 5y x = 5 2 y + 4 3y = 2x y = 2 ( 5 y + 4) y = 5y y = 5y Combine like terms and solve for y 20 = 2y y = Plug answer for y into the isolated expression you found in step 2. Note: you can substitute y=-10 into any equation with x and y, but this equation is the easiest since its already solved for x x = 5 2 y + 4 x = 5 2 ( 10) + 4 x = x = 21 Final Answer: (x, y) = ( 21, 10)

10 10 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Trigonometry Trigonometry deals with right triangles. It can be used to find unknown sides or angles. Knowing the values of two sides of a right triangle, or of one side and one angle, is enough to solve for all the remaining information. sinθ = opp hyp adj cosθ = hyp opp tanθ = adj θ = sin 1 ( opp hyp ) = cos 1 ( adj hyp ) = tan 1 ( opp adj ) a 2 + b 2 = c 2 Example: Find the unknown side and the angle. Pythagorean theorem to find x x 2 = 18 2 Finding the angle Another option for finding x x= θ = cos 1 ( 7 18 ) = sin (67.11) = x 18 x = In physics, we use trigonometry to combine vectors. Vectors are quantities like displacement that have a length and a direction. For example, if you walk 10.0m West and 4.0m North, how far are you from the starting point? If we represent 10.0 West and 4.0 North as vector arrows, we can make a triangle to answer this. The hypotenuse is an arrow that connects our starting point and our ending point, and its length will tell us exactly how far apart those points are. The angle at the base of our triangle will also tell you which direction you traveled in. 4.0m N 10.0m West θ The hypotenuse of the triangle is 10.8m long, and the angle is This tells us that to go right from our start to end point, we could walk exactly 10.8m in a direction that is North of West

11 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 11 Multiple vectors can be combined using a special form of addition called vector addition. Vector addition takes into account not just the magnitude (how long) of the vector, but also the direction. There are 5 basic steps to vector addition: 1. Check to see if any vectors you need to add are at an angle. If so, break them into components. 2. Combine all the horizontal pieces, adding when they point in the same direction and subtracting when they are opposite. This gives you your net horizontal. 3. Combine all the vertical pieces, adding when they point in the same direction and subtracting when hey are opposite. This gives you your net vertical. 4. Draw a new right triangle using your net horizontal and net vertical as the legs. 5. Use Pythagorean theorem to find the hypotenuse of the triangle and trigonometry to find the angle. Ex: Combine the following vectors: 20m 35m Both of these vectors are at an angle, so they will both need to be broken down into components. 20m m Net horizontal: Combine East and West. Since they point opposite directions, subtract: = 3.03 East Net vertical: Combine 9.08 North and North. Since they point in the same direction, add: = North Put net components together into final triangle: Answer: 40.91m, N of E Use Pythagorean Theorem and inverse tangent to find magnitude and direction of final vector.

12 12 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Units, Conversions & Measurement: To accurately complete an experiment it is important to be able to take accurate measurements. In general, when taking a measurement you want the last decimal place to be an estimate. In the ruler to the right, the most exact marking are at one-tenth of a centimeter. This means that you can estimate one decimal place farther, to the hundredths place. A measurement you might record for the arrow would be 2.78cm. You definitely know the 2 and the 0.7cm because those are marked on the ruler, and you are estimating that it is 0.08cm past the 2.7 mark. The metric system is the standard measurement system in science, and with good reason. Larger and smaller units are all based on powers of ten, which greatly simplifies conversions. This is in contrast to the standard system where units are chosen based on arbitrary definitions and require extensive memorization to convert. In physics, the following are the standard units used for measurements: Length Meter (m) Mass Kilogram (kg) Time Second (s) Speed Meter per second (m/s) You should know and be able to convert between the metric prefixes below: Prefix Symbol Multiple of Base tera- T There are meters in one Terameter giga- G 10 9 There are 10 9 m in one Gm mega- M 10 6 There are 10 6 m in one Mm kilo- k 10 3 There are 1,000 m in one km Base unit (meter, gram, liter, second ) deci- d 10-1 There are 10 decimeters in one meter centi- c 10-2 There are 100 cm in one m milli- m 10-3 There are 1,000 mm in one m micro- μ 10-6 There are 10 6 μm in one m nano- n 10-9 There are 10 9 nm in one m pico- p There are pm in one m Big units Small units The multiple/fraction number is how many base units there are in one unit with a prefix. For example, there are 1,000 meters in a kilometer and grams in a milligram. Conversion bridges are useful to convert between prefixes, as shown below. Conversion bridges are based on the idea that you are multiplying by a number equivalent to one. The numerator and denominator of the fraction must be equivalent to each other, just in different units.

13 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 13 Example: How many micrometers are in 2.8 decimeters? 2.8 dm 0.1m 1 dm 1 μm 10 6 m = 280,000 μm = 2.8x105 μm or 1m 2.8 dm 10 dm 106 μm 1m = 280,000 μm = 2.8x105 μm or (using difference in powers of ten) 2.8 dm 105 μm 1dm = 280,000 μm = 2.8x105 μm Conversion bridges can also be used to convert non-metric units if you are given a conversion factor. Example: Convert 60 miles per hour to m/s, given that 1 mile = 1609 m. 60 miles 1 hour 1609m 1 mile 1 hour 60 minutes 1 minute 60 seconds = 26.8 m s Some units for volume can be the cube of length units, for example cm 3. When converting these units, the conversion factor itself must be cubed because we only know the conversion between linear (first-power) length units, not cubed volume units. Example: A car engine has a displacement of 3.5L. What is this displacement in cubic inches? 1 ml = 1cm 3, 1 in =2.54cm 3.5L 1000 ml 1L 1cm3 1mL ( 1 in cm ) = 213.6in 3 The inch to cm conversion needed to be cubed because we only know the conversion between linear inches and linear cm, not cubic inches and cubic cm.

14 14 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Practice Problems Directions: Solve the following problems, being sure to show all work. 1.) 16.7 kilograms is how many grams? 2.) 560 nm is how many meters? 3.) 15 years is how many seconds? 4.) seconds is how many years? 5.) m/s is how many kilometers per hour? 6. In Europe, fuel mileage is measured in liters / 100km. Convert 30 miles per gallon to L/100km.

15 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 15 Trigonometry Review 7. Using the properties of triangles, prove that A C in the drawing below. Directions: Use the figure below to answer problems Simplify as much as you can. a b c 8) Find c if given a and b. 9.) Find a if given b and c. 10.) Find a if given c and. 11.) Find b if given a and. 12.) Find c if given b and. 13.) Find if given b and c. 14.) Find if given a and b. 15.) If a = 2.0 and c = 7.0, what is b? 16.) If c = 10.0 and = 60, what is b? 17.) If a = 12.0 and = 30, what is b?

16 16 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Trigonometry and Adding Vectors: Combine the following vectors. Be sure to give a magnitude (number) and a direction. 18.) 80m South and 20m West 19.) 40m North, 10m South, 30m East 20.) 25m at 45 0 North of East, 18m East 21.) 150m at 30 0 South of West, 200m at 60 0 South of East 22.) 25N at 30 0 S of W, 34N at 70 0 S of W 23.) 12 m/s North, 16 m/s East, (N stands for Newtons, a measure of force) 15 m/s at 45 0 S of W. 24.) An airplane flies North at 200 m/s, and encounters a wind blowing at 30 m/s at 35 0 W of N. a) What is the resultant velocity of the airplane? b) How far is the plane from its starting point after one hour?

17 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 17 Algebra Review Directions: Solve the following equations for the given variable and conditions. Simplify if needed. Example: 2x + xy = z. Solve for x. x(2 + y) = z x = z 2 + y 25.) v + v = Solve for v1. 26.) a = v. Solve for t. t 27.) v 2 f = v 2 i + 2ad A.) Solve for vi. B.) Solve for d. 28.) d f = d i + v o t at 2 A.) Solve for vo. B.) Solve for t, if vo = 0. C.) Solve for t, if di = df.

18 18 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 29.) F = m v f - v i t f - t i A.) Solve for vf, if ti = 0. B.) Solve for tf, if vf = 0 and ti = ) a c = v 2 r. Solve for v. 31.) mgsinq = mmgcosq. Solve for θ. 32.) 1 2 mv 2 f + mgh f = 1 2 mv 2 i + mgh i A.) Solve for hf, if hi = 0 and vf = 0. B.) Solve for vf, if hf = ) Ft = mv f - mv i. Solve for vf. 34.) m v + m v = (m + m )v. Solve for vi,2. 1 i,1 2 i,2 1 2 f 35.) m v + m v = m v + m v 1 i,1 2 i,2 1 f,1 2 f,2. Solve for vf,2 if vi,1 = 0.

19 36.) (F sinq)r + (-F sinf)r = 0. Solve for r2. AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW ) -kx + m(-g) = 0. Solve for m. 38.) F g = G m 1m 2 r 2. Solve for r. 39.) L - Lcosq = v2 2 Solve for L. 40.) mv 2 R = G Mm. Solve for v. 2 R 41.) T = 2p L g. Solve for g. 42.) 1 R T = 1 R R 2. Solve for RT. 43.) 1 2 mv 2 f kx 2 = 1 2 mv 2 i + mgh i. Solve for x if vf = 0.

20 20 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Miscellaneous Directions: Simplify without using a calculator. Remember to show all of your work. 44._) ) ) Proportions: 47.) Consider z = x s2, c = ab, l = m - n, or r =. Answer with increases/decreases: 2 y t a.) As x increases and y stays constant, z. b.) As y increases and x stays constant, z. c.) As x increases and z stays constant, y. d.) If y is tripled and z stays constant, x is multiplied by. e.) If x is tripled and y is doubled, z is multiplied by. f.) If z is doubled and y is tripled, x is multiplied by. g.) As a increases and c stays constant, b. h.) As c increases and b stays constant, a. i.) As b increases and a stays constant, c. j.) If a and b are both doubled, c is multiplied by. k.) If a and c are both doubled, c is multiplied by. l.) As n increases and m stays constant, l. m.) As l increases and n stays constant, m. n.) If s is tripled and t stays constant, r is multiplied by. o.) If t is doubled and s stays constant, r is multiplied by. p.) If r and s are both doubled, t is multiplied by. q.) If r is doubled and t is cut in half, t is multiplied by.

21 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 21 Systems of equations: Use the equations in each problem to solve for the specified variable in the given terms. Simplify as far as possible. 48.) x 10 = 5t and x = 2t + 3t 2. Solve for x and t numerically. 49.) F f = mf N and F N = mgcosq. Solve for in terms of Ff, m, g, and. 50.) F 1 + F 2 = F T and F 1 d 1 = F 2 d 2. Solve for F1 in terms of FT, d1, and d2. 51.) F c = ma c and a c = v 2. Solve for r in terms of Fc, m, and v. r 52.) T = 2p L g and T = 1. Solve for L in terms of, g, and f. f

22 22 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Graphing Equations 53.) If r = c x*t was graphed on an r vs. t graph, what would the following be? Slope: y-intercept: 54.) On the y vs. x graphs below, sketch the relationships given. a.) y = mx + b, if m > 0 and b = 0. b.) y = mx + b, if m < 0 and b > 0. c.) y = x 2 d.) y = 5x 2 e.) y = x f.) y = 5 x g.) y = 1 x h.) y = 5 x

23 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 23 For each of the following graphs, find (a) The proportion / relationship between y and x, and (b) the numerical equation of the line. To complete (b), you will need to identify at least three data points and re-graph as a linear function. The graphs may have negative slopes or y-intercepts

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25 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 25 Marbles in Cylinder Lab Complete the following lab analysis: You received a graduated cylinder with three identical marbles and an unknown amount of water already in it. You placed extra identical marbles in the cylinder and obtained the data below. Use the data to graph a bestfit line showing the relationship between the water level and the number of marbles. The y- intercept should be visible on the graph. Label your axes and include units. From the graph, determine a mathematical formula for the water level for any number of marbles. Lastly, give an explanation of your formula in words. Make sure to give an explanation of the slope and y-intercept of your formula. Number of Marbles Water level (ml) in Water Graph below: Formula: Explanation of the formula in words: (Include the meaning of the slope and y-intercept.)

26 26 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW Position vs Time Lab Complete the following lab analysis: You record the position of your car at even time intervals. Use the data to find the exact mathematical relationship between position and time. The y-intercept should be visible on the graph. Label your axes and include units. From the graph, determine a mathematical formula for the position at any time. Lastly, give an explanation of your formula in words. Make sure to give an explanation of the slope and y-intercept of your formula. Time (s) Position (m) Formula: Explanation of the formula in words: (Include the units and meaning of the slope and y- intercept.)

27 AP PHYSICS 1 // MATH & SCIENCE SKILLS REVIEW 27 Current vs. Resistance Lab Complete the following lab analysis: You record the current flowing through a wire for several different resistors. Use the data to find the exact mathematical relationship between current and resistance. The y-intercept should be visible on the graph. Label your axes and include units. From the graph, determine a mathematical formula for the current for any resistance. Lastly, give an explanation of your formula in words. Make sure to give an explanation of the slope and y-intercept of your formula. Resistance (Ohms) Current (Amps) Formula: Explanation of the formula in words: (Include the units of the slope and y-intercept.)