Chapter 1

The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in \((a)\) years, \((b)\) seconds.

(1) The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in \((a)\) years, \((b)\) seconds.

How many significant figures do each of the following numbers have: \((a) 214,(b) 81.60,\) (c) \(7.03,\) (d) 0.03 , (e) \(0.0086,(f)\) 3236, and (g) \(8700 ?\)

(1) How many significant figures do each of the following numbers have: $$ 214, \text { (b) } 81.60, \text { (c) } 7.03, \text { (d) } 0.03 $$ $$ (e) 0.0086,(f) 3236, \text { and }(g) 8700 ? $$

(I) Write the following numbers in powers of ten notation: (a) \(1.156,(b) 21.8,(c) 0.0068\) (d) \(328.65,(e) 0.219,\) and \((f) 444\)

$$ \begin{array}{l}{\text { (1) Write the following numbers in powers of ten notation: }} \\ {\text { (a) } 1.156,(b) 21.8,(c) 0.0068,(d) 328.65,(e) 0.219, \text { and }(f) 444 \text { . }}\end{array} $$

(1) Write out the following numbers in full with the correct number of zeros: (a) \(8.69 \times 10^{4}\), (b) \(9.1 \times 10^{3}\), (c) \(8.8 \times 10^{-1},\) (d) \(4.76 \times 10^{2}\), and (e) \(3.62 \times 10^{-5}\)

What is the percent uncertainty in the measurement \(5.48 \pm 0.25 \mathrm{~m} ?\)

(II) What is the percent uncertainty in the measurement 5.48\(\pm 0.25 \mathrm{m} ?\)

Time intervals measured with a stopwatch typically have an uncertainty of about \(0.2 \mathrm{~s}\), due to human reaction time at the start and stop moments. What is the percent uncertainty of a hand-timed measurement of \((a) 5 \mathrm{~s},(b) 50 \mathrm{~s},(c) 5 \mathrm{~min} ?\)

(II) Time intervals measured with a stopwatch typically have an uncertainty of about 0.2 s, due to human reaction time at the start and stop moments. What is the percent uncertainty of a hand-timed measurement of $$ (a) 5 s,(b) 50 s,(c) 5 \min ? $$

Add \(\left(9.2 \times 10^{3} \mathrm{~s}\right)+\left(8.3 \times 10^{4} \mathrm{~s}\right)+\left(0.008 \times 10^{6} \mathrm{~s}\right)\).

(II) Add $$ \left(9.2 \times 10^{3} s\right)+\left(8.3 \times 10^{4} s\right)+\left(0.008 \times 10^{6} s\right) $$

Multiply \(2.079 \times 10^{2} \mathrm{~m}\) by \(0.082 \times 10^{-1}\), taking into account significant figures.

$$ \begin{array}{l}{\text { (II) Multiply } 2079 \times 10^{2} \mathrm{m} \text { by } 0.082 \times 10^{-1} \text { , taking into }} \\ {\text { account significant figures. }}\end{array} $$

For small angles \(\theta\), the numerical value of \(\sin \theta\) is approximately the same as the numerical value of \(\tan \theta .\) Find the largest angle for which sine and tangent agree to within two significant figures.

$$ \begin{array}{l}{\text { (III) For small angles } \theta \text { , the numerical value of } \sin \theta \text { is }} \\ {\text { approximately the same as the numerical value of tan } \theta \text { . }} \\ {\text { Find the largest angle for which sine and tangent agree to }} \\ {\text { within two significant figures. }}\end{array} $$

What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is \(r=0.84 \pm 0.04 \mathrm{~m} ?\)

$$ \begin{array}{l}{\text { (III) What, roughly, is the percent uncertainty in the volume }} \\ {\text { of a spherical beach ball whose radius is } r=0.84 \pm 0.04 \mathrm{m} \text { ? }}\end{array} $$

Write the following as full (decimal) numbers with standard units: \((a) 286.6 \mathrm{~mm},(b) 85 \mu \mathrm{V},(c) 760 \mathrm{mg},(d) 60.0 \mathrm{ps}\) (e) \(22.5 \mathrm{fm},(f)\) 2.50 gigavolts.

$$ \begin{array}{l}{\text { (1) Write the following as full (decimal) numbers with stan- }} \\ {\text { dard units: }(a) 286.6 \mathrm{mm},(b) 85 \mu \mathrm{V},(c) 760 \mathrm{mg},(d) 60.0 \mathrm{ps}} \\ {(e) 22.5 \mathrm{fm},(f) 2.50 \text { gigavolts. }}\end{array} $$

Determine your own height in meters, and your mass in kg.

$$ \begin{array}{l}{\text { 13. (I) } \quad \text { Determine }} \\ {\text { your own height in }} \\ {\text { meters, and your mass }} \\ {\text { in } \mathrm{kg}}\end{array} $$

The Sun, on average, is 93 million miles from Earth. How many meters is this? Express (a) using powers of ten, and (b) using a metric prefix.

(I) The Sun, on average, is 93 million miles from Earth. How many meters is this? Express \((a) \quad\) using powers of ten, and \((b)\) using a metric prefix.

What is the conversion factor between \((a) \mathrm{ft}^{2}\) and \(\mathrm{yd}^{2}\) (b) \(\mathrm{m}^{2}\) and \(\mathrm{ft}^{2}\) ?

An airplane travels at \(950 \mathrm{~km} / \mathrm{h}\). How long does it take to travel \(1.00 \mathrm{~km} ?\)

(II) An Airplane travels at 950 \(\mathrm{km} / \mathrm{h}\) . How long does it take to travel 1.00 \(\mathrm{km} ?\)

A typical atom has a diameter of about \(1.0 \times 10^{-10} \mathrm{~m}\). (a) What is this in inches? (b) Approximately how many atoms are there along a \(1.0-\mathrm{cm}\) line?

(II) A typical atom has a diameter of about \(1.0 \times 10^{-10} \mathrm{m}\) (a) What is this in inches? (b) Approximately how many atoms are there along a 1.0 -cm line?

18\. (II) Express the following sum with the correct number of significant figures: \(1.80 \mathrm{m}+\) \(142.5 \mathrm{cm}+5.34 \times 10^{5} \mu \mathrm{m} .\)

Determine the conversion factor between (a) \(\mathrm{km} / \mathrm{h}\) and \(\mathrm{mi} / \mathrm{h},(b) \mathrm{m} / \mathrm{s}\) and \(\mathrm{ft} / \mathrm{s},\) and \((c) \mathrm{km} / \mathrm{h}\) and \(\mathrm{m} / \mathrm{s}\).

(II) Determine the conversion factor between \((a) \mathrm{km} / \mathrm{h}\) and \(\mathrm{mi} / \mathrm{h},(b) \mathrm{m} / \mathrm{s}\) and \(\mathrm{ft} / \mathrm{s},\) and \((c) \mathrm{km} / \mathrm{h}\) and \(\mathrm{m} / \mathrm{s}\) .

How much longer (percentage) is a one-mile race than a \(1500-\mathrm{m}\) race ("the metric mile")?

(II) How much longer (percentage) is a one-mile race than a 1500 -m race ("the metric mile")?

A light-year is the distance light travels in one year (at speed \(\left.=2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\). (a) How many meters are there in 1.00 light-year? (b) An astronomical unit (AU) is the average distance from the Sun to Earth, \(1.50 \times 10^{8} \mathrm{~km} .\) How many AU are there in 1.00 light-year? (c) What is the speed of light in \(\mathrm{AU} / \mathrm{h} ?\)

$$ \begin{array}{l}{\text { (II) A light-year is the distance light travels in one year }} \\ {\text { (at speed }=2.998 \times 10^{8} \mathrm{m} / \mathrm{s} ) \text { . (a) How many meters are }} \\ {\text { there in } 1.00 \text { light-year? }(b) \text { An astronomical unit }(\mathrm{AU}) \text { is }}\end{array} $$ $$ \begin{array}{l}{\text { the average distance from the Sun to Earth, } 1.50 \times 10^{8} \mathrm{km} \text { . }} \\ {\text { How many } \mathrm{AU} \text { are there in } 1.00 \text { light-year? (c) What is the }} \\ {\text { speed of light in } \mathrm{AU} / \mathrm{h} \text { ? }}\end{array} $$

If you used only a keyboard to enter data, how many years would it take to fill up the hard drive in your computer that can store 82 gigabytes \(\left(82 \times 10^{9}\right.\) bytes \()\) of data? Assume "normal" eight-hour working days, and that one byte is required to store one keyboard character, and that you can type 180 characters per minute.

$$\begin{array}{l}{\text { (II) If you used only a keyboard to enter data, how many }} \\ {\text { years would it take to fill up the hard drive in your }} \\\ {\text { computer that can store } 82 \text { gigabytes }\left(82 \times 10^{9} \text { bytes) of }\right.}\end{array} $$ data? Assume "normal" cight- hour working days, and that one byte is required to store one keyboard character, and that you can type 180 characters per minute.

Estimate the order of magnitude (power of ten) of: \((a) 2800\), (b) \(86.30 \times 10^{2}\) (c) \(0.0076,\) and \((d) 15.0 \times 10^{8}\)

(1) Estimate the order of magnitude (power of ten) of: \((a) 2800\) , (b) \(86.30 \times 10^{2},(c) 0.0076,\) and \((d) 15.0 \times 10^{8} .\)

You are in a hot air balloon, \(200 \mathrm{~m}\) above the flat Texas plains. You look out toward the horizon. How far out can you see-that is, how far is your horizon? The Earth's radius is about \(6400 \mathrm{~km}\).

(III) You are in a hot air balloon, 200 \(\mathrm{m}\) above the flat Texas plains. You look out toward the horizon. How far out can you sec-that is, how far is your horizon? The Earth's radius is about 6400 \(\mathrm{km}\) .

I agree to hire you for 30 days and you can decide between two possible methods of payment: either (1) \(\$ 1000\) a day, or (2) one penny on the first day, two pennies on the second day and continue to double your daily pay each day up to day 30 . Use quick estimation to make your decision, and justify it.

(III) I agree to hire you for 30 days and you can decide between two possible methods of payment: cither \((1) \$ 1000\) a day, or (2) one penny on the first day, two pennics on the scoond day and continue to double your daily pay cach day up to day 30. Use quick estimation to make your decision, and justify it.

What are the dimensions of density, which is mass per volume?

(1) What are the dimensions of density, which is mass per volume?

The speed \(v\) of an object is given by the equation \(v=A t^{3}-B t,\) where \(t\) refers to time. \((a)\) What are the dimensions of \(A\) and \(B ?(b)\) What are the SI units for the constants \(A\) and \(B ?\)

$$ \begin{array}{l}{\text { (II) The spced } v \text { of an object is given by the equation }} \\ {v=A t^{3}-B t, \text { where } t \text { refers to time. (a) What are the }} \\ {\text { dimensions of } A \text { and } B ?(b) \text { What are the SI units for the }} \\ {\text { constants } A \text { and } B ?}\end{array} $$

Three students derive the following equations in which \(x\) refers to distance traveled, \(v\) the speed, \(a\) the acceleration \(\left(\mathrm{m} / \mathrm{s}^{2}\right), t\) the time, and the subscript zero (0) means a quantity at time \(t=0:(a) x=v t^{2}+2 a t,(b) x=v_{0} t+\frac{1}{2} a t^{2},\) and (c) \(x=v_{0} t+2 a t^{2}\). Which of these could possibly be correct according to a dimensional check?