OpenStax University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
Page 0 of 16
,OpenStax University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
Conceptual Questions
1. What is physics?
Solution
Physics is the science concerned with describing the interactions of energy, matter, space, and
time to uncover the fundamental mechanisms that underlie every phenomenon.
2. Some have described physics as a “search for simplicity.” Explain why this might be an
appropriate description.
Solution
Physics searches for a small number of theories that explain all natural phenomena.
3. If two different theories describe experimental observations equally well, can one be said to be
more valid than the other (assuming both use accepted rules of logic)?
Solution
No, neither of these two theories is more valid than the other. Experimentation is the ultimate
decider. If experimental evidence does not suggest one theory over the other, then both are
equally valid. A given physicist might prefer one theory over another on the grounds that one
seems more simple, more natural, or more beautiful than the other, but that physicist would
quickly acknowledge that he or she cannot say the other theory is invalid. Rather, he or she
would be honest about the fact that more experimental evidence is needed to determine which
theory is a better description of nature.
4. What determines the validity of a theory?
Solution
Repeated experimental tests of its implications by various groups of experimenters determines
the validity of a theory.
5. Certain criteria must be satisfied if a measurement or observation is to be believed. Will the
criteria necessarily be as strict for an expected result as for an unexpected result?
Solution
Probably not. As the saying goes, “Extraordinary claims require extraordinary evidence.”
6. Can the validity of a model be limited or must it be universally valid? How does this compare
with the required validity of a theory or a law?
Solution
Models are meant to capture only some aspects of a physical system. A theory or law should
describe all aspects of the physical systems within their domain of applicability.
7. Identify some advantages of metric units.
Solution
Conversions between units require factors of 10 only, which simplifies calculations. Also, the
same basic units can be scaled up or down using metric prefixes to sizes appropriate for the
problem at hand.
8. What are the SI base units of length, mass, and time?
Solution
The SI base unit of length is the meter (m). The SI base unit of mass is the kilogram (kg). The SI
base unit of time is the second (s).
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Unit 1: Mechanics
Chapter 1: Units and Measurement
9. What is the difference between a base unit and a derived unit? (b) What is the difference
between a base quantity and a derived quantity? (c) What is the difference between a base
quantity and a base unit?
Solution
a. Base units are defined by a particular process of measuring a base quantity whereas derived
units are defined as algebraic combinations of base units. b. A base quantity is chosen by
convention and practical considerations. Derived quantities are expressed as algebraic
combinations of base quantities. c. A base unit is a standard for expressing the measurement of a
base quantity within a particular system of units. So, a measurement of a base quantity could be
expressed in terms of a base unit in any system of units using the same base quantities. For
example, length is a base quantity in both SI and the English system, but the meter is a base unit
in the SI system only.
10. For each of the following scenarios, refer to the following figure and table to determine
which metric prefix on the meter is most appropriate for each of the following scenarios. (a) You
want to tabulate the mean distance from the Sun for each planet in the solar system. (b) You
want to compare the sizes of some common viruses to design a mechanical filter capable of
blocking the pathogenic ones. (c) You want to list the diameters of all the elements on the
periodic table. (d) You want to list the distances to all the stars that have now received any radio
broadcasts sent from Earth 10 years ago.
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Chapter 1: Units and Measurement
Prefix Symbol Meaning Prefix Symbol Meaning
yotta- Y 1024 yocto- y 10–24
zetta- Z 1021 zepto- z 10–21
exa- E 1018 atto- a 10–18
peta- P 1015 femto- f 10–15
tera- T 1012 pico- p 10–12
giga- G 109 nano- n 10–9
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Chapter 1: Units and Measurement
mega- M 106 micro- 10–6
kilo- k 103 milli- m 10–3
hecto- h 102 centi- c 10–2
deka- da 101 deci- d 10–1
Solution
Answers may vary. a. All planets are closer to the Sun than the solar system diameter, which is
of the order 1013 m, so using terameters (1012 m) makes sense for this scenario.; b. Virus
diameters are of the order 10–7 m, so either micrometers (10–6 m) or nanometers (10–9 m) make
sense for this scenario.; c. All the atoms in the periodic table are at least as big as hydrogen,
which has a diameter of order 10–10 m, so either nanometers (10–9 m) or picometers (10—12 m)
make sense for this scenario.; d. The stars that have received our broadcasts from 10 years ago
are within 10 light-years (of order 1017 m) of Earth. So, either petameters (1015 m) or exameters
(1018 m) make sense for this scenario.
11. (a) What is the relationship between the precision and the uncertainty of a measurement? (b)
What is the relationship between the accuracy and the discrepancy of a measurement?
Solution
a. Uncertainty is a quantitative measure of precision. b. Discrepancy is a quantitative measure of
accuracy.
12. What information do you need to choose which equation or equations to use to solve a
problem?
Solution
You need to know which physical quantities you were given (either explicitly or implicitly) and
which physical quantities you need to find.
13. What should you do after obtaining a numerical answer when solving a problem?
Solution
Check to make sure it makes sense and assess its significance.
Problems
14. Find the order of magnitude of the following physical quantities. (a) The mass of Earth’s
atmosphere: kg; (b) The mass of the Moon’s atmosphere: 25,000 kg; (c) The mass of
Earth’s hydrosphere: kg; (d) The mass of Earth: kg; (e) The mass of the
Moon: kg; (f) The Earth–Moon distance (semimajor axis): m; (g) The
mean Earth–Sun distance: m; (h) The equatorial radius of Earth: m; (i) The
mass of an electron: kg; (j) The mass of a proton: kg; (k) The mass of
the Sun: kg.
Solution
a. 1019 kg; b. 104 kg; c. 1021 kg; d. 1025 kg; e. 1023 kg; f. 109 m; g. 1011 m; h. 107 m; i. 10–30 kg; j.
10–27 kg; k. 1030 kg
15. Use the orders of magnitude you found in the previous problem to answer the following
questions to within an order of magnitude. (a) How many electrons would it take to equal the
mass of a proton? (b) How many Earths would it take to equal the mass of the Sun? (c) How
many Earth–Moon distances would it take to cover the distance from Earth to the Sun? (d) How
many Moon atmospheres would it take to equal the mass of Earth’s atmosphere? (e) How many
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Unit 1: Mechanics
Chapter 1: Units and Measurement
moons would it take to equal the mass of Earth? (f) How many protons would it take to equal the
mass of the Sun?
Solution
a. 10–27/10–30 = 103; b. 1030/1025 = 105; c. 1011/109 = 102; d. 1019/104 = 1015; e. 1025/1023 = 102; f.
1030/10–27 = 1057
For the remaining questions, you need to use the following figure to obtain the necessary orders
of magnitude of lengths, masses, and times.
16. Roughly how many heartbeats are there in a lifetime?
Solution
109 heartbeats
17. A generation is about one-third of a lifetime. Approximately how many generations have
passed since the year 0?
Solution
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Chapter 1: Units and Measurement
102 generations
18. Roughly how many times longer than the mean life of an extremely unstable atomic nucleus
is the lifetime of a human?
Solution
1031 times
19. Calculate the approximate number of atoms in a bacterium. Assume the average mass of an
atom in the bacterium is 10 times the mass of a proton.
Solution
1011 atoms
20. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is
10 times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a
human?
Solution
a. 1012 cells per hummingbird b. 1016 cells per person
21. Assuming one nerve impulse must end before another can begin, what is the maximum firing
rate of a nerve in impulses per second?
Solution
103 nerve impulses/s
22. About how many floating-point operations can a supercomputer perform each year?
Solution
107/10–17 = 1024 floating-point operations/yr
23. Roughly how many floating-point operations can a supercomputer perform in a human
lifetime?
Solution
109/10–17 = 1026 floating-point operations per human lifetime
24. The following times are given using metric prefixes on the base SI unit of time: the second.
Rewrite them in scientific notation without the prefix. For example, 47 Ts would be rewritten as
s. (a) 980 Ps; (b) 980 fs; (c) 17 ns; (d)
Solution
a. s; b. s; c. s; d. s
25. The following times are given in seconds. Use metric prefixes to rewrite them so the
numerical value is greater than one but less than 1000. For example, s could be written
as either 7.9 cs or 79 ms. (a) s; (b) 0.045 s; (c) s; (d) s.
Solution
a. 957 ks; b. 4.5 cs or 45 ms; c. 550 ns; d. 31.6 Ms
26. The following lengths are given using metric prefixes on the base SI unit of length: the
meter. Rewrite them in scientific notation without the prefix. For example, 4.2 Pm would be
rewritten as m. (a) 89 Tm; (b) 89 pm; (c) 711 mm; (d)
Solution
a. m; b. m; c. m; d. m
27. The following lengths are given in meters. Use metric prefixes to rewrite them so the
numerical value is bigger than one but less than 1000. For example, m could be written
either as 7.9 cm or 79 mm. (a) m; (b) 0.0074 m; (c) m; (d) m.
Solution
a. 75.9 Mm; b. 7.4 mm; c. 88 pm; d. 16.3 Tm
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Chapter 1: Units and Measurement
28. The following masses are written using metric prefixes on the gram. Rewrite them in
scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40 Mg would
be written as kg. (a) 23 mg; (b) 320 Tg; (c) 42 ng; (d) 7 g; (e) 9 Pg.
Solution
a. kg; b. kg; c. kg; d. kg; e. kg
29. The following masses are given in kilograms. Use metric prefixes on the gram to rewrite
them so the numerical value is bigger than one but less than 1000. For example, kg could
be written as 70 cg or 700 mg. (a) kg; (b) kg; (c) kg; (d) kg;
(e) kg.
Solution
a. 3.8 cg or 38 mg; b. 230 Eg; c. 24 ng; d. 8 Eg e. 4.2 g
30. The volume of Earth is on the order of 1021 m3. (a) What is this in cubic kilometers (km3)? (b)
What is it in cubic miles (mi3)? (c) What is it in cubic centimeters (cm3)?
Solution
a. 1012 km3; b. 1011 mi3; c. 1027 cm3
31. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters
per second? (b) How many miles per hour is this?
Solution
a. 27.8 m/s; b. 62 mi/h
32. A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it
exceeding the 90 km/h speed limit?
Solution
a. ; b. At 120 km/h, the car is exceeding the speed limit.
33. In SI units, speeds are measured in meters per second (m/s). But, depending on where you
live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per
hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h
or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show
that (a) and (b) .
Solution
a. km/h; b. mi/h
34. American football is played on a 100-yd-long field, excluding the end zones. How long is the
field in meters? (Assume that 1 m = 3.281 ft.)
Solution
91.4 m
35. Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its
area in square feet? (Assume that 1 m = 3.281 ft.)
Solution
36. What is the height in meters of a person who is 6 ft 1.0 in. tall?
Solution
1.85 m
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Chapter 1: Units and Measurement
37. Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers?
(Assume that 1 m = 3.281 ft.)
Solution
8.847 km
38. The speed of sound is measured to be 342 m/s on a certain day. What is this measurement in
kilometers per hour?
Solution
39. Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate
has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b)
What is its speed in kilometers per million years?
Solution
a. ; b. 40 km/My
40. The average distance between Earth and the Sun is m. (a) Calculate the average
speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in
miles per hour?
Solution
a. ; b.
41. The density of nuclear matter is about 1018 kg/m3. Given that 1 mL is equal in volume to cm3,
what is the density of nuclear matter in megagrams per microliter (that is, )?
Solution
42. The density of aluminum is 2.7 g/cm3. What is the density in kilograms per cubic meter?
Solution
43. A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm,
where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?
Solution
62.4 lbm/ft3
44. A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to
millimeters per second.
Solution
0.166 mm/s
45. It takes radians (rad) to get around a circle, which is the same as 360°. How many
radians are in 1°?
Solution
0.017 rad
46. Light travels a distance of about m/s. A light-minute is the distance light travels in 1
min. If the Sun is m from Earth, how far away is it in light-minutes?
Solution
8 light-minutes
47. A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.
Solution
1 light-nanosecond
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Chapter 1: Units and Measurement
48. An electron has a mass of kg. A proton has a mass of What is
the mass of a proton in electron-masses?
Solution
electron-masses
49. A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic
meters?
Solution
m3
50. A student is trying to remember some formulas from geometry. In what follows, assume
is area, is volume, and all other variables are lengths. Determine which formulas are
dimensionally consistent. (a) ; (b) ; (c) ; (d) ; (e)
.
Solution
a. This formula is dimensionally consistent; both terms have dimensions L3. b. This formula is
dimensionally consistent; all three terms have dimensions L2. c. This formula is not
dimensionally consistent. d. This formula is not dimensionally consistent. e. This formula is
dimensionally consistent; both terms have dimensions L3.
51. Consider the physical quantities s, v, a, and t with dimensions , , ,
and . Determine whether each of the following equations is dimensionally consistent. (a)
; (b) ; (c) ; (d) .
Solution
a. Yes, both terms have dimension L2T-2 b. No. c. Yes, both terms have dimension LT-1 d. Yes,
both terms have dimension LT-2
52. Consider the physical quantities , , , , , and with dimensions [m] = M, [s] = L, [v]
= LT–1, [a] = LT–2, [t] = T, and [r] = L. Assuming each of the following equations is
dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation:
(a) F = ma; (b) K = 0.5mv2; (c) p = mv; (d) W = mas; (e) L = mvr.
Solution
a. [F] = MLT–2; b. [K] = ML2T–2; c. [p] = MLT–1; d. [W] = ML2T–2; e. [L] = ML2T–1
53. Suppose quantity is a length and quantity is a time. Suppose the quantities and are
defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of
the quantity a? What are the dimensions of (c) , (d) , and (e) da/dt?
Solution
a. [v] = LT–1; b. [a] = LT–2; c. = L; d. LT–1; e. LT–3
54. Suppose [V] = L3, [ ] = ML–3, and [t] = T. (a) What is the dimension of ? (b) What
is the dimension of dV/dt? (c) What is the dimension of
Solution
a. M; b. L3T–1; c. MT–1
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Unit 1: Mechanics
Chapter 1: Units and Measurement
Page 0 of 16
,OpenStax University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
Conceptual Questions
1. What is physics?
Solution
Physics is the science concerned with describing the interactions of energy, matter, space, and
time to uncover the fundamental mechanisms that underlie every phenomenon.
2. Some have described physics as a “search for simplicity.” Explain why this might be an
appropriate description.
Solution
Physics searches for a small number of theories that explain all natural phenomena.
3. If two different theories describe experimental observations equally well, can one be said to be
more valid than the other (assuming both use accepted rules of logic)?
Solution
No, neither of these two theories is more valid than the other. Experimentation is the ultimate
decider. If experimental evidence does not suggest one theory over the other, then both are
equally valid. A given physicist might prefer one theory over another on the grounds that one
seems more simple, more natural, or more beautiful than the other, but that physicist would
quickly acknowledge that he or she cannot say the other theory is invalid. Rather, he or she
would be honest about the fact that more experimental evidence is needed to determine which
theory is a better description of nature.
4. What determines the validity of a theory?
Solution
Repeated experimental tests of its implications by various groups of experimenters determines
the validity of a theory.
5. Certain criteria must be satisfied if a measurement or observation is to be believed. Will the
criteria necessarily be as strict for an expected result as for an unexpected result?
Solution
Probably not. As the saying goes, “Extraordinary claims require extraordinary evidence.”
6. Can the validity of a model be limited or must it be universally valid? How does this compare
with the required validity of a theory or a law?
Solution
Models are meant to capture only some aspects of a physical system. A theory or law should
describe all aspects of the physical systems within their domain of applicability.
7. Identify some advantages of metric units.
Solution
Conversions between units require factors of 10 only, which simplifies calculations. Also, the
same basic units can be scaled up or down using metric prefixes to sizes appropriate for the
problem at hand.
8. What are the SI base units of length, mass, and time?
Solution
The SI base unit of length is the meter (m). The SI base unit of mass is the kilogram (kg). The SI
base unit of time is the second (s).
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Chapter 1: Units and Measurement
9. What is the difference between a base unit and a derived unit? (b) What is the difference
between a base quantity and a derived quantity? (c) What is the difference between a base
quantity and a base unit?
Solution
a. Base units are defined by a particular process of measuring a base quantity whereas derived
units are defined as algebraic combinations of base units. b. A base quantity is chosen by
convention and practical considerations. Derived quantities are expressed as algebraic
combinations of base quantities. c. A base unit is a standard for expressing the measurement of a
base quantity within a particular system of units. So, a measurement of a base quantity could be
expressed in terms of a base unit in any system of units using the same base quantities. For
example, length is a base quantity in both SI and the English system, but the meter is a base unit
in the SI system only.
10. For each of the following scenarios, refer to the following figure and table to determine
which metric prefix on the meter is most appropriate for each of the following scenarios. (a) You
want to tabulate the mean distance from the Sun for each planet in the solar system. (b) You
want to compare the sizes of some common viruses to design a mechanical filter capable of
blocking the pathogenic ones. (c) You want to list the diameters of all the elements on the
periodic table. (d) You want to list the distances to all the stars that have now received any radio
broadcasts sent from Earth 10 years ago.
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Unit 1: Mechanics
Chapter 1: Units and Measurement
Prefix Symbol Meaning Prefix Symbol Meaning
yotta- Y 1024 yocto- y 10–24
zetta- Z 1021 zepto- z 10–21
exa- E 1018 atto- a 10–18
peta- P 1015 femto- f 10–15
tera- T 1012 pico- p 10–12
giga- G 109 nano- n 10–9
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Chapter 1: Units and Measurement
mega- M 106 micro- 10–6
kilo- k 103 milli- m 10–3
hecto- h 102 centi- c 10–2
deka- da 101 deci- d 10–1
Solution
Answers may vary. a. All planets are closer to the Sun than the solar system diameter, which is
of the order 1013 m, so using terameters (1012 m) makes sense for this scenario.; b. Virus
diameters are of the order 10–7 m, so either micrometers (10–6 m) or nanometers (10–9 m) make
sense for this scenario.; c. All the atoms in the periodic table are at least as big as hydrogen,
which has a diameter of order 10–10 m, so either nanometers (10–9 m) or picometers (10—12 m)
make sense for this scenario.; d. The stars that have received our broadcasts from 10 years ago
are within 10 light-years (of order 1017 m) of Earth. So, either petameters (1015 m) or exameters
(1018 m) make sense for this scenario.
11. (a) What is the relationship between the precision and the uncertainty of a measurement? (b)
What is the relationship between the accuracy and the discrepancy of a measurement?
Solution
a. Uncertainty is a quantitative measure of precision. b. Discrepancy is a quantitative measure of
accuracy.
12. What information do you need to choose which equation or equations to use to solve a
problem?
Solution
You need to know which physical quantities you were given (either explicitly or implicitly) and
which physical quantities you need to find.
13. What should you do after obtaining a numerical answer when solving a problem?
Solution
Check to make sure it makes sense and assess its significance.
Problems
14. Find the order of magnitude of the following physical quantities. (a) The mass of Earth’s
atmosphere: kg; (b) The mass of the Moon’s atmosphere: 25,000 kg; (c) The mass of
Earth’s hydrosphere: kg; (d) The mass of Earth: kg; (e) The mass of the
Moon: kg; (f) The Earth–Moon distance (semimajor axis): m; (g) The
mean Earth–Sun distance: m; (h) The equatorial radius of Earth: m; (i) The
mass of an electron: kg; (j) The mass of a proton: kg; (k) The mass of
the Sun: kg.
Solution
a. 1019 kg; b. 104 kg; c. 1021 kg; d. 1025 kg; e. 1023 kg; f. 109 m; g. 1011 m; h. 107 m; i. 10–30 kg; j.
10–27 kg; k. 1030 kg
15. Use the orders of magnitude you found in the previous problem to answer the following
questions to within an order of magnitude. (a) How many electrons would it take to equal the
mass of a proton? (b) How many Earths would it take to equal the mass of the Sun? (c) How
many Earth–Moon distances would it take to cover the distance from Earth to the Sun? (d) How
many Moon atmospheres would it take to equal the mass of Earth’s atmosphere? (e) How many
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Chapter 1: Units and Measurement
moons would it take to equal the mass of Earth? (f) How many protons would it take to equal the
mass of the Sun?
Solution
a. 10–27/10–30 = 103; b. 1030/1025 = 105; c. 1011/109 = 102; d. 1019/104 = 1015; e. 1025/1023 = 102; f.
1030/10–27 = 1057
For the remaining questions, you need to use the following figure to obtain the necessary orders
of magnitude of lengths, masses, and times.
16. Roughly how many heartbeats are there in a lifetime?
Solution
109 heartbeats
17. A generation is about one-third of a lifetime. Approximately how many generations have
passed since the year 0?
Solution
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102 generations
18. Roughly how many times longer than the mean life of an extremely unstable atomic nucleus
is the lifetime of a human?
Solution
1031 times
19. Calculate the approximate number of atoms in a bacterium. Assume the average mass of an
atom in the bacterium is 10 times the mass of a proton.
Solution
1011 atoms
20. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is
10 times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a
human?
Solution
a. 1012 cells per hummingbird b. 1016 cells per person
21. Assuming one nerve impulse must end before another can begin, what is the maximum firing
rate of a nerve in impulses per second?
Solution
103 nerve impulses/s
22. About how many floating-point operations can a supercomputer perform each year?
Solution
107/10–17 = 1024 floating-point operations/yr
23. Roughly how many floating-point operations can a supercomputer perform in a human
lifetime?
Solution
109/10–17 = 1026 floating-point operations per human lifetime
24. The following times are given using metric prefixes on the base SI unit of time: the second.
Rewrite them in scientific notation without the prefix. For example, 47 Ts would be rewritten as
s. (a) 980 Ps; (b) 980 fs; (c) 17 ns; (d)
Solution
a. s; b. s; c. s; d. s
25. The following times are given in seconds. Use metric prefixes to rewrite them so the
numerical value is greater than one but less than 1000. For example, s could be written
as either 7.9 cs or 79 ms. (a) s; (b) 0.045 s; (c) s; (d) s.
Solution
a. 957 ks; b. 4.5 cs or 45 ms; c. 550 ns; d. 31.6 Ms
26. The following lengths are given using metric prefixes on the base SI unit of length: the
meter. Rewrite them in scientific notation without the prefix. For example, 4.2 Pm would be
rewritten as m. (a) 89 Tm; (b) 89 pm; (c) 711 mm; (d)
Solution
a. m; b. m; c. m; d. m
27. The following lengths are given in meters. Use metric prefixes to rewrite them so the
numerical value is bigger than one but less than 1000. For example, m could be written
either as 7.9 cm or 79 mm. (a) m; (b) 0.0074 m; (c) m; (d) m.
Solution
a. 75.9 Mm; b. 7.4 mm; c. 88 pm; d. 16.3 Tm
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28. The following masses are written using metric prefixes on the gram. Rewrite them in
scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40 Mg would
be written as kg. (a) 23 mg; (b) 320 Tg; (c) 42 ng; (d) 7 g; (e) 9 Pg.
Solution
a. kg; b. kg; c. kg; d. kg; e. kg
29. The following masses are given in kilograms. Use metric prefixes on the gram to rewrite
them so the numerical value is bigger than one but less than 1000. For example, kg could
be written as 70 cg or 700 mg. (a) kg; (b) kg; (c) kg; (d) kg;
(e) kg.
Solution
a. 3.8 cg or 38 mg; b. 230 Eg; c. 24 ng; d. 8 Eg e. 4.2 g
30. The volume of Earth is on the order of 1021 m3. (a) What is this in cubic kilometers (km3)? (b)
What is it in cubic miles (mi3)? (c) What is it in cubic centimeters (cm3)?
Solution
a. 1012 km3; b. 1011 mi3; c. 1027 cm3
31. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters
per second? (b) How many miles per hour is this?
Solution
a. 27.8 m/s; b. 62 mi/h
32. A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it
exceeding the 90 km/h speed limit?
Solution
a. ; b. At 120 km/h, the car is exceeding the speed limit.
33. In SI units, speeds are measured in meters per second (m/s). But, depending on where you
live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per
hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h
or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show
that (a) and (b) .
Solution
a. km/h; b. mi/h
34. American football is played on a 100-yd-long field, excluding the end zones. How long is the
field in meters? (Assume that 1 m = 3.281 ft.)
Solution
91.4 m
35. Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its
area in square feet? (Assume that 1 m = 3.281 ft.)
Solution
36. What is the height in meters of a person who is 6 ft 1.0 in. tall?
Solution
1.85 m
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,OpenStax University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
37. Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers?
(Assume that 1 m = 3.281 ft.)
Solution
8.847 km
38. The speed of sound is measured to be 342 m/s on a certain day. What is this measurement in
kilometers per hour?
Solution
39. Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate
has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b)
What is its speed in kilometers per million years?
Solution
a. ; b. 40 km/My
40. The average distance between Earth and the Sun is m. (a) Calculate the average
speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in
miles per hour?
Solution
a. ; b.
41. The density of nuclear matter is about 1018 kg/m3. Given that 1 mL is equal in volume to cm3,
what is the density of nuclear matter in megagrams per microliter (that is, )?
Solution
42. The density of aluminum is 2.7 g/cm3. What is the density in kilograms per cubic meter?
Solution
43. A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm,
where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?
Solution
62.4 lbm/ft3
44. A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to
millimeters per second.
Solution
0.166 mm/s
45. It takes radians (rad) to get around a circle, which is the same as 360°. How many
radians are in 1°?
Solution
0.017 rad
46. Light travels a distance of about m/s. A light-minute is the distance light travels in 1
min. If the Sun is m from Earth, how far away is it in light-minutes?
Solution
8 light-minutes
47. A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.
Solution
1 light-nanosecond
Page 8 of 16
, OpenStax University Physics Volume I
Unit 1: Mechanics
Chapter 1: Units and Measurement
48. An electron has a mass of kg. A proton has a mass of What is
the mass of a proton in electron-masses?
Solution
electron-masses
49. A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic
meters?
Solution
m3
50. A student is trying to remember some formulas from geometry. In what follows, assume
is area, is volume, and all other variables are lengths. Determine which formulas are
dimensionally consistent. (a) ; (b) ; (c) ; (d) ; (e)
.
Solution
a. This formula is dimensionally consistent; both terms have dimensions L3. b. This formula is
dimensionally consistent; all three terms have dimensions L2. c. This formula is not
dimensionally consistent. d. This formula is not dimensionally consistent. e. This formula is
dimensionally consistent; both terms have dimensions L3.
51. Consider the physical quantities s, v, a, and t with dimensions , , ,
and . Determine whether each of the following equations is dimensionally consistent. (a)
; (b) ; (c) ; (d) .
Solution
a. Yes, both terms have dimension L2T-2 b. No. c. Yes, both terms have dimension LT-1 d. Yes,
both terms have dimension LT-2
52. Consider the physical quantities , , , , , and with dimensions [m] = M, [s] = L, [v]
= LT–1, [a] = LT–2, [t] = T, and [r] = L. Assuming each of the following equations is
dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation:
(a) F = ma; (b) K = 0.5mv2; (c) p = mv; (d) W = mas; (e) L = mvr.
Solution
a. [F] = MLT–2; b. [K] = ML2T–2; c. [p] = MLT–1; d. [W] = ML2T–2; e. [L] = ML2T–1
53. Suppose quantity is a length and quantity is a time. Suppose the quantities and are
defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of
the quantity a? What are the dimensions of (c) , (d) , and (e) da/dt?
Solution
a. [v] = LT–1; b. [a] = LT–2; c. = L; d. LT–1; e. LT–3
54. Suppose [V] = L3, [ ] = ML–3, and [t] = T. (a) What is the dimension of ? (b) What
is the dimension of dV/dt? (c) What is the dimension of
Solution
a. M; b. L3T–1; c. MT–1
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