Chapter 1

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

$$ \begin{array}{l}{\text { (II) Show that the following combination of the three funda- }} \\ {\text { mental constants of nature that we used in Example } 10 \text { of }} \\ {\text { "Introduction, Measurement, Estimating" (that is } G, c, \text { and } h} \\ {\text { forms a quantity with the dimensions of time: }}\end{array} $$ \(t_{\mathrm{P}}=\sqrt{\frac{G h}{c^{5}}}\) This quantity, \(t_{1}\) , is called the Planck time and is thought to be the carliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

Global positioning satellites (GPS) can be used to determine positions with great accuracy. If one of the satellites is at a distance of \(20,000 \mathrm{~km}\) from you, what percent uncertainty in the distance does a \(2-\mathrm{m}\) uncertainty represent? How many significant figures are needed in the distance?

(a) How many seconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? \((c)\) How many years are there in 1.00 second?

(a) How many scconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? (c) How many years are there in 1.00 second?

American football uses a field that is 100 yd long, whereas a regulation soccer field is \(100 \mathrm{~m}\) long. Which field is longer, and by how much (give yards, meters, and percent)?

One hectare is defined as \(1.000 \times 10^{4} \mathrm{~m}^{2}\). One acre is \(4.356 \times 10^{4} \mathrm{ft}^{2} .\) How many acres are in one hectare?

An average family of four uses roughly \(1200 \mathrm{~L}\) (about 300 gallons) of water per day \(\left(1 \mathrm{~L}=1000 \mathrm{~cm}^{3}\right)\). How much depth would a lake lose per year if it uniformly covered an area of \(50 \mathrm{~km}^{2}\) and supplied a local town with a population of 40,000 people? Consider only population uses, and neglect evaporation and so on.

$$ \begin{array}{l}{\text { An average family of four uses roughly } 1200 \mathrm{L} \text { (about) }} \\ {300 \text { gallons) of water per day }\left(1 \mathrm{L}=1000 \mathrm{cm}^{3}\right) . \text { How much }} \\\ {\text { depth would a lake lose per year if it uniformly covered an }} \\\ {\text { arca of } 50 \mathrm{km}^{2} \text { and supplicd a local town with a population }} \\ {\text { of } 40,000 \text { people? Consider only population uses, and }} \\ {\text { neglect evaporation and so on. }}\end{array} $$

How big is a ton? That is, what is the volume of something that weighs a ton? To be specific, estimate the diameter of a 1 -ton rock, but first make a wild guess: will it be \(1 \mathrm{ft}\) across, \(3 \mathrm{ft},\) or the size of a car? [Hint: Rock has mass per volume about 3 times that of water, which is \(1 \mathrm{~kg}\) per liter \(\left(10^{3} \mathrm{~cm}^{3}\right)\) or 62 lb per cubic foot.

A certain audio compact disc (CD) contains 783.216 megabytes of digital information. Each byte consists of exactly 8 bits. When played, a CD player reads the CD's digital information at a constant rate of 1.4 megabits per second. How many minutes does it take the player to read the entire CD?

A certain audio compact dise (CD) contains 783.216 megabytes of digital information. Each byte consists of exactly 8 bits. When played, a CD player reads the CD's digital information at a constant rate of 1.4 megabits per scoond. How many minutes does it take the player to read the entire CD?

A heavy rainstorm dumps \(1.0 \mathrm{~cm}\) of rain on a city \(5 \mathrm{~km}\) wide and \(8 \mathrm{~km}\) long in a \(2-\mathrm{h}\) period. How many metric tons \(\left(1\right.\) metric ton \(\left.=10^{3} \mathrm{~kg}\right)\) of water fell on the city? \(\left(1 \mathrm{~cm}^{3}\right.\) of water has a mass of \(1 \mathrm{~g}=10^{-3} \mathrm{~kg}\).) How many gallons of water was this?

$$ \begin{array}{l}{\text { A heavy rainstorm dumps } 1.0 \mathrm{cm} \text { of rain on a city } 5 \mathrm{km} \text { wide }} \\ {\text { and } 8 \mathrm{km} \text { long in a } 2 \text { -h period. How many metric tons }} \\\ {\left(1 \text { metric ton }=10^{3} \mathrm{kg}\right) \text { of water fell on the city? }\left(1 \mathrm{cm}^{3} \text { of }\right.} \\ {\text { water has a mass of } 1 \mathrm{g}=10^{-3} \mathrm{kg} . \text { ) How many gallons }} \\ {\text { of water was this? }}\end{array} $$

Estimate how many days it would take to walk around the world, assuming \(10 \mathrm{~h}\) walking per day at \(4 \mathrm{~km} / \mathrm{h}\).

One liter \(\left(1000 \mathrm{~cm}^{3}\right)\) of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just one molecule thick, with adjacent molecules just touching. estimate the diameter of the oil slick. Assume the oil molecules have a diameter of \(2 \times 10^{-10} \mathrm{~m}\)

Jean camps beside a wide river and wonders how wide it is She spots a large rock on the bank directly across from her. She then walks upstream until she judges that the angle between her and the rock, which she can still sce clearly, is now at an angle of \(30^{\circ}\) downstream (Fig. 16). Jean measures her stride to be about 1 yard long. The distance back to her camp is 120 strides. About how yards and in meters, yards and in meters, is the river?

A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate is this watch, expressed as a percentage?

The diameter of the Moon is \(3480 \mathrm{~km}\). What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth?

Determine the percent uncertainty in \(\theta\), and in \(\sin \theta\), when (a) \(\theta=15.0^{\circ} \pm 0.5^{\circ}\) (b) \(\theta=75.0^{\circ} \pm 0.5^{\circ}\).

Determine the peroent uncertainty in \(\theta,\) and in \(\sin \theta,\) when (a) \(\theta=15.0^{\circ} \pm 0.5^{\circ},(b) \quad \theta=75.0^{\circ} \pm 0.5^{\circ} .\)

If you began walking along one of Earth's lines of longitude and walked north until you had changed latitude by 1 minute of arc (there are 60 minutes per degree), how far would you have walked (in miles)? This distance is called a

Make a rough estimate of the volume of your body (in \(\mathrm{m}^{3}\) ).

\text { Make a rough estimate of the volume of your body (in } \mathrm{m}^{3} \text { ). }

The American Lung Association gives the following formula for an average person's expected lung capacity \(V\) (in liters, $$ \begin{array}{l} \text { where } 1 \mathrm{~L}=10^{3} \mathrm{~cm}^{3} \text { ): } \\ \qquad V=4.1 \mathrm{H}-0.018 \mathrm{~A}-2.69 \end{array} $$ where \(H\) and \(A\) are the person's height (in meters), and age (in years), respectively. In this formula, what are the units of the numbers \(4.1,0.018,\) and \(2.69 ?\)

The density of an object is defined as its mass divided by its volume. Suppose the mass and volume of a rock are measured to be \(8 \mathrm{~g}\) and \(2.8325 \mathrm{~cm}^{3}\). To the correct number of significant figures, determine the rock's density.

$$ \begin{array}{l}{\text { The density of an object is defined as its mass divided by its }} \\ {\text { volume. Suppose the mass and volume of a rock are }} \\ {\text { measured to be } 8 \mathrm{g} \text { and } 2.8325 \mathrm{cm}^{3} . \text { To the correct number }} \\ {\text { of significant figures, determine the rock's density. }}\end{array} $$

Recent findings in astrophysics suggest that the observable Universe can be modeled as a sphere of radius \(R=13.7 \times 10^{9}\) light-years with an average mass density of about \(1 \times 10^{-26} \mathrm{~kg} / \mathrm{m}^{3},\) where only about \(4 \%\) of the Universe's total mass is due to "ordinary" matter (such as protons, neutrons, and electrons). Use this information to estimate the total mass of ordinary matter in the observable Universe. (1 light-year \(\left.=9.46 \times 10^{15} \mathrm{~m} .\right)\)