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IB Physics HL Year 1 First Assignment
Welcome to Mrs. Terzella’s IB Physics HL year 1 course! I am thrilled that you are here and I am fully prepared to take you on a journey to learn concepts that are central to physics and formulas that illuminate these concepts. I hope that this proves to be the best class that you have taken. Expect to be challenged! This is a college level course where you will be using your knowledge and understanding of everything you have learned in all of your classes to solve problems, analyze situations, arrange materials, compare data, design labs, and build incredible things. You cannot expect to acquire the understanding you need to do well on the IB Exam by merely attending class and listening to the teacher. You have to become INVOLVED. You have to PARTICIPATE. If you get stuck, see ME, or other students. Ask for HELP. Your classmates will be your new best friends. You must study regularly. Students who study regularly have a good foundation to build on for new topics. This will pay off! If you are unorganized or inconsistent, things may start to fall apart ­ and nobody wants that to happen. Busy work is not assigned in this course so do what I ask you to do regularly! Especially the homework!! This summer assignment is intended to introduce you to the math you will be using throughout next year in IB Physics, so that these will not need to be reviewed during the school year and the focus can remain on the new material. Most of these techniques/concepts will not be new to you (I hope!), but sometimes they are used differently in physics than they are in math. Each section of this assignment will contain a brief explanation of the concept, followed by a sample problem, followed by practice problems. My intent is that you will use this (and the answer key that will be on Edline) as a reference throughout the year. ​
Do all your work on a separate piece of paper, and keep the packet itself. The paper will be due the Friday of the first week of school. Topic 1: Fractions Don’t laugh – it’s easy to forget the rules for fractions when you haven’t used them in a while. To multiply fractions, simply multiply the numerators to get the numerator of the answer, and multiply the denominators to get the denominator of the answer. Reduce to lowest terms. Ex.: 6/7 x 4/9 = 24/63 = ​
8/21 To divide fractions, take the reciprocal of the second fraction then multiply them. Ex.: 6/7 ÷ 4/9 = 6/7 x 9/4 = 54/28 = ​
27/14 To add or subtract fractions, you need to first find the least common denominator. Convert all the fractions to have that denominator, then add or subtract the numerators. Ex.: 6/7 + 4/9 ⇒ LCD = 63 ⇒ 54/63 + 28/63 = ​
82/63 Problems: 1. 87/5 x 1/7 2. 3/5 ÷ 2/3 3. 1/6 + 1/4 4. 7/88 – 3/2 5. 2/9 + 8/9 Topic 2: Solving Equations A ​
linear or first­order equation​
is one in which all the terms are raised to the first power. If this type of equation is graphed, the result will be a straight line. To solve this type of equation, combine like terms if necessary then isolate the variable using addition, subtraction, multiplication, or division. Ex.: 14x + 3.97 = 24.1 ­ 5(6x + 3.1) ⇒ 14x + 3.97 = 24.1 ­ 30x ­ 15.5 ⇒ 14x + 3.97 = ­30x + 8.6 ⇒ 44x = 4.63 ⇒ ​
x = 0.105 Problems: 6. 5(2x + 9) = 7x ­ 62.7 7. 24 = 33.8/x ­ 5.03 8. 8/x + 5/(4x) = 1/7 9. 15 ­ x/8 = ­1 10. (x ­ 14)/(­2x + 55) = 17 ​
A ​
quadratic equation​
is a polynomial of the form ​
ax2​
​
+ bx + c = 0​
(or one that can be rearranged into that form). These can be solved by factoring, by completing the square, or using the quadratic formula. Since physics uses real­world data in the problems (instead of conveniently selected numbers) they will very rarely be able to be factored. You will almost always need to use the quadratic formula: ​ ​
−b±​
√​
b​−4ac x ​
=​
2​
2a Note that this will give you two solutions for x, so you will use the context of the problem to determine the correct solution. Ex.:
2​
2t​
– 4t – 3 = 0 ⇒ Problems: 2​
11. x​
+ x ­ 3.75 = 0 2​
12. x​
­ 14x + 45 = 0 2​
13. 0.3x​
+ 0.15x ­ 11.7 = 0 2​
14. x​
­ 1.375x + 0.375 = 0 2​
15. 16x​
– 8x ­ 3 = 0 An equation that has two or more variables in it cannot be solved (or, it has an infinite number of solutions). To solve for the variables, two or more ​
simultaneous equations are needed. If you have the same number of equations as you have variables, the equations can be solved. There are three ways of doing this: graphing, elimination, or substitution. In physics problems, substitution is usually the best option. Solve one equation for one of the variables, then substitute the result into the other equation. Repeat until you have a numerical answer for one of the variables. Ex.:
2.9x – 17y = 33
and
0.4x/y = 217 ⇒ 0.4x = 217y ⇒ x = 542.5y ⇒ put this into the first equation: 2.9(542.5y) – 17y = 33 ⇒ 1573.25y – 17y = 33 ⇒ 1556.25y = 33 ⇒ ​
y = 0.021 Don’t forget to find the second variable, using your result for the first: x = 542.5(0.021) = ​
11.39 = x Problems: 16. 220 = xy and 17. 3x + 17y = 52 and 0 = ­9.8x + 0.7y 0.2x – 3.8y = 3.2 18. x + y – z = 0 and 5x – 6y + z = 8 19. 15/x = 2y and 34x = 3y + 9.9x 20. (2+x)/3y = 4 and (5 – y)/(2+x) = 0.88 and­4x + 35y + 9z = 12 Topic 3: Scientific Notation When numbers are too large or too small to write conveniently, scientific notation is used. A number written in scientific notation has two parts: a number between 1 and 10 followed by an exponent of 10. The exponent represents how many places to move the decimal point in the first number. If it is positive, the decimal moves to the right, and if it negative, the decimal moves to the left. Ex.:
4​
7.75 x 10​
= 77500 ­8​
2.9 x 10​
= 0.000000029 Problems: 6
21. 8.03 x 10​
­8
22. 3.55 x 10​
3
23. 5.90 x 10​
24. 0.00000276 25. 704300000000 Topic 4: Metric Prefixes The metric system is based on 10, and uses prefixes to denote different multiples of base units. Whichever base unit is being used (meters, grams, hertz, etc.) the same prefixes are used. Some prefixes are commonly used in physics, and some are not. The ones you will be expected to know are: Name Abbreviation Meaning 9 giga G 10​
mega M 6 10​
kilo k 3 10​
­​
base​
­ 0 10​
centi c ­2 10​
milli m ­3 10​
micro µ ­6 10​
nano n ­9 10​
Within the metric system, converting from one unit to another is done by moving the decimal point. There are many different ways to remember which direction and how many places to move the decimal point, one of which is using the exponent notation. To do this, first subtract the exponents for each unit. The answer tells you how many places to move the decimal point. If the answer is positive, move the decimal to the right, and if it’s negative, move the decimal to the left. Ex.:
Convert 45cm into Gm. ­2​
9 Centi: 10​
Giga: 10​
­2 – 9 = ­11 ⇒ The decimal should move 11 places to the left: 45cm = ​
0.00000000045Gm Problems: 26. Convert 3C to µC. 27. Convert 84Mm to km. 28. Convert 0.05kg to cg. 29. Convert 700nm to m. 30. Convert 1300mL to kL. Topic 5: Dimensional Analysis If you are converting outside the metric system, you need one or more ​
conversion factors. ​
Starting with the original quantity you want to convert, you will multiply or divide by these conversion factors until your answer is written in the units required by the problem. One strategy for setting up these problems is to write each conversion factor as a fraction. This can be done because a conversion factor, such as 1ft = 12in, consists of two equal quantities. When you express them as a fraction, the fraction is equal to 1, so it can be multiplied by the original number without changing the number. Arrange your expression so that units you need to eliminate will cancel out, then multiply. Ex.: Convert 4 years into seconds. Conversion factors: 1yr = 365 days 1day = 24hr 1hr = 3600s 8​
4yr x ​
365days​
x ​
24hr​
x ​
3600s​
= ​
1.26x10​
s 1yr 1day 1hr Problems: 31. Convert 144 miles into cm. 32. Convert 35 mph into m/s. 33. Your car's gas tank holds 18.6 gallons and is one quarter full. Your car gets ​
16 miles/gal. You see a sign saying, "Next gas 73 miles." Your brother, who Is driving, is sure you'll make it without running out of gas. Is he right? 34.
How much bleach, in mL, would you need to make a quart of 5 percent bleach solution? 35.
A box measures 3.12 ft in length, 0.0455 yd in width and 7.87 inches in height. What is its volume in cubic centimeters? Topic 6: Significant Figures In any experiment, your final result is only as good as your least accurate measurement. In a lab report, you should always account for this in a thoughtful error analysis, but you will also account for this using significant figures. The rules to determine which digits are significant and which are not are as follows: ●
Non­zero numbers are always significant. ●
Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures. ●
Zeros that do nothing but set the decimal point are ​
not significant. Thus, 470,000 has two significant figures. ●
Trailing zeros that aren't needed to hold the decimal point ​
are significant. For example, 4.00 has three significant figures. ●
If you aren't sure whether or not a zero is significant, cover it up and see if the number would have the same value without it. If not, then it is ​
not​
significant (it's just there to set the decimal point.) The exception to this is a single zero before a decimal point, such as the zero in 0.55. That zero is not significant. ●
When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement. ●
When measurements are added or subtracted, the answer can contain no more significant figures after the decimal point than the least accurate measurement. Ex.:
53.00 + 207.3 = 260.30 → ​
260.3 53.00 x 207.3 = 10,986.9 → ​
10,990 Sometimes the only way to accurately report an answer to the right number of significant figures is to use scientific notation. For example, if I wanted to write 10,986.9 to three 4​
significant figures, I would have to write 1.10x10​
­ writing 10,900 would be rounding off incorrectly, and writing 11,000 would only have two significant figures. Problems: 36. 14.039 + 2.00 37. 75 x 0.8 38. 200 + 0.115 39. 3,040/95.00 40. 84.90 ­ 16.4 Topic 7: Percent Error A way to determine how much error there is in an experiment (and then account for it in an error analysis) is to calculate the ​
percent error​
. If you have an accepted value for a number you have calculated/measured (such as the acceleration due to gravity, the speed of sound in air, etc.), the formula for percent error is: % error = ​
| accepted value ­ experimental value |​
x 100 accepted value If you are simply comparing two values that should be the same (such as two different measurements for the frequency of a wave or the velocity of a ball) then the quantity you calculate is called the ​
percent difference​
and there is no "accepted value." In that case, the numerator of the formula will remain the same (subtract your two numbers and take the absolute value) but instead of dividing by the accepted value, divide by the average of your two numbers. Problems: 2​
41. The acceleration due to gravity (​
g​
) on Earth is 9.81m/s​
. You perform an 2​
experiment and get a value for ​
g​
of 9.75m/s​
. Find your percent error. 42. You launch a ball off a table five times and measure the distance it travels horizontally. Your results are: 2.66m, 2.46m, 2.72m, 2.70m, and 2.50m. Find the percent difference between your highest and lowest result. 43. The stripes printed on a resistor indicate that its resistance is 620Ω, plus or minus 5%. What are the highest and lowest possible values for its resistance? 44. Find the percent difference between 12.03 and 12.02. 8​
45. The accepted value for the speed of light (​
c​
) is 3x10​
m/s. In 1676 Ole Roemer did an experiment that yielded a value for ​
c​
of 220,000,000m/s. What was his percent error? Topic 8: Vectors A vector is any quantity that has both a size and a direction. Most often, these are written as a ​
magnitude​
, including units of measurement, (such as 7.7m or 0.59N) and an ​
angle​
. In physics class, all angles are measured in ​
degrees​
. If no other reference points are mentioned in a given problem, the angle can be assumed to be the angle to the positive x­axis. Sometimes a different angle will be given and you will first have to find the angle to the positive x­axis. Vectors can be ​
resolved​
into their x­ and y­components using sine and cosine. This is where always using the angle to the positive x­axis is beneficial: the horizontal, or x­component ​
will always be associated with​
cosine​
, and the vertical, or​
y­component will always be associated with ​
sine​
. This makes them easier to remember. The formulas you use to find the components are simply the definitions of sine and cosine: A​
x = Acosø ​
A​
y = Asinø ​
o​
Ex.: A bird flies at 13m/s at an angle of 35​
to the ground. Find the components of the bird's motion. V​
x = 13cos35 = 11m/s ​
V​
y = 13sin35 = 7.5m/s ​
The magnitude and direction of a vector can also be determined if the x­ and y­components are known, using the Pythagorean Theorem to find the magnitude, and the definition of tangent to find the direction: 2​
2​
2
A​
= A​
tanø = A​
/A​
x​ + A​
y​
y​
x Positive and negative signs are primarily used to denote direction. Horizontally, anything directed ​
left​
is ​
negative​
, and anything directed ​
right​
is ​
positive​
. Vertically, anything directed ​
up​
is ​
positive​
and anything directed ​
down​
is ​
negative​
. (Picture an xy axis and the signs of points in each of the four quadrants.) Ex.: A boy climbed a set of stairs that had a covered a horizontal distance of 1.66m while rising to a height of 2.02m. What is the overall displacement (distance and direction) of the boy from his starting point? 2​
2​
2​
A​
= 1.66​
+ 2.02 ​
= 6.84 ⇒ ​
A = 2.62m o tanø = 2.02/1.66 = 1.23 ⇒ ​
ø = 50.9​
Remember, drawing diagrams is always useful and often essential! Problems: o​
46.
A cannonball is launched from a cannon at 46.03m/s at an angle of 45​
. What are the components of its motion? o​
47. Find the components of a vector that is 298 units long at an angle of 3​
. 48.
A bug on a wall crawls 12cm to the right, then 7cm down. Find the angle and size of its displacement. 49.
Two ropes attached to a heavy box exert forces of 300N to the left, and 450N up. What is the size and direction of the total force on the box? o​
50.
A plane flies overhead at angle of 14​
to the ground. You watch as its shadow moves under it, and you notice that its shadow travels a distance of 203m. How far did the plane actually travel? How much height did it gain? Topic 9: Graphing Many of the labs we will perform will require graphs. Sometimes those graphs will be just to show a relationship you already knew, sometimes they will be used to determine the relationship between two variables, and sometimes they will be used to determine the value of a quantity. Though we will sometimes graph by hand, most of the time we will be using Excel, Google Sheets, or Plot.ly. You should be able to do the following: ●
make a scatterplot with the correct quantities on the x­ and y­axes ●
change the title of the graph ●
label the x­ and y­axes ●
add a trendline ●
force the trendline to have a certain y­intercept (such as (0,0)) ●
display the equation of the trendline ●
modify the equation of the trendline to use the correct variables (instead of just x and y) If you are not sure how to do any of these things, use the Help function in the various spreadsheet programs, look it up online, or simply play around until you figure it out. Problem: Graph the following data using a spreadsheet program. Put voltage (​
V,​
measured in Volts) on the y­axis and current (​
I,​
measured in Amperes) on the x­axis. On your graph, do all of the following: ●
title the graph "Ohm's Law" ●
label the x­ and y­axes so that they match the column labels on the data table ● add a linear trendline Current (A)
Voltage (V) 0.00 0 0.014 3 0.027 6 0.044 9 0.054 12 0.070 15 0.081 18 0.095 21 0.11 24 0.12 27 0.15 30 0.15 33 0.16 36 0.18 39 0.19 42 0.20 45 0.22 48 0.23 51 0.24 54 0.26 57