College Physics 2013

EST A water molecule in a glass of water has $10^{14}$ collisions with other molecules every second. Estimate the number of years a college football player would have to play $(24 \text { hours }$ a day) to have the same number of collisions. Explain all of your assumptions.

What are the molar masses of molecular and atomic hydrogen, helium, oxygen, and nitrogen? What are their molecular masses?

EST Estimate the number of hydrogen atoms in the Sun. The mass of the Sun is $2 \times 10^{30} \mathrm{kg}$ . About 70$\%$ of it by mass is hydrogen, 30$\%$ is helium, and there is a negligible amount of other elements.

The average particle density in the Milky Way galaxy is about one particle per cubic centimeter. Express this number in SI units $\left(\mathrm{kg} / \mathrm{m}^{3}\right)$ . Indicate any assumptions you made.

(a) What is the concentration (number per cubic meter) of the molecules in air at normal conditions? (b) What is the average distance between molecules compared to the dimensions of the molecules? (c) Can you consider air to be an ideal gas? Explain your answer.

EST Estimate the number of collisions that one molecule of air in the physics classroom experiences every second. List all the assumptions that you made.

What is the mass of a water molecule in kilograms? What is the mass of an average air molecule in kilograms?

You find that the average gauge pressure in your car tires is about 35 psi. How many newtons per square meter is it? What is gauge pressure?

BIO Forced vital capacity Physicians use a machine called a spirometer to measure the maximum amount of air a person can exhale (called the forced vital capacity). Suppose you can exhale 4.8 L. How many kilograms of air do you exhale? What assumptions did you make to answer the question? How do
these assumptions affect the result?

A container is at rest with respect to a desk. Inside the container a particle is moving horizontally at a speed v with respect to the desk. It collides with a vertical wall of the container elastically and rebounds. Qualitatively, determine the direction and the speed of the particle if the wall is (a) at rest with respect to the desk; (b) moving in the same direction as the particle at a speed smaller than the particle’s; and (c)
moving in the direction opposite to the motion of the particle at a smaller speed.

Hitting tennis balls against a wall $\mathrm{A} 0.058$ -kg tennis ball, traveling at 25 $\mathrm{m} / \mathrm{s}$ , hits a wall, rebounds with the same speed in the opposite direction, and is hit again by another player, causing the ball to return to the wall at the same speed. The ball returns to the wall once every 0.60 $\mathrm{s}$ . (a) Determine the force that the ball exerts on the wall averaged over the time between collisions. State the assumptions that you made. (b) If 10 people are practicing against a wall with an area of 30 $\mathrm{m}^{2}$ , what is the average pressure of the 10 tennis balls against the wall?

Friends throw snowballs at the wall of a 3.0 $\mathrm{m} \times 6.0 \mathrm{m}$ barn. The snowballs have mass 0.10 $\mathrm{kg}$ and hit the wall moving at an average speed of 6.0 $\mathrm{m} / \mathrm{s}$ . They do not rebound. Determine the average pressure exerted by the snowballs on the
molecules of an ideal gas hitting the walls of their container?

A ball moving at a speed of 3.0 $\mathrm{m} / \mathrm{s}$ with respect to the ground hits a stationary wall at a $30^{\circ}$ angle with respect to the surface of the wall. Determine the direction and the magnitude of the velocity of the ball after it rebounds. Explain carefully what physics principles you used to find the answer. What assumptions did you make? How will the answer change if one or more of them are not valid?

Oxygen tank for mountains Consider an oxygen tank for a mountain climbing trip. The mass of one molecule of oxygen is $5.3 \times 10^{-26} \mathrm{kg}$ . What is the pressure that oxygen exerts on the inside walls of the tank if its concentration is $10^{25}$ particles/m $^{3}$ and its rms speed is 600 $\mathrm{m} / \mathrm{s} ?$ What assumptions did you make?

You have five molecules with the following speeds: 300 $\mathrm{m} / \mathrm{s}$ , $400 \mathrm{m} / \mathrm{s}, 500 \mathrm{m} / \mathrm{s}, 450 \mathrm{m} / \mathrm{s},$ and 550 $\mathrm{m} / \mathrm{s} .$ What is their average speed?

What is the rms speed of the molecules in the previous problem? Is it different from the average speed?

Two gases in different containers have the same concentration and same rms speed. The mass of a molecule of the first gas is twice the mass of a molecule of the second gas. What can you say about their pressures? Explain.

You are hiking up a mountain. About halfway up you pass through a cloud and become moist from cloud water. How can this water be at such a high elevation? To answer this question, compare the molar masses and densities of dry air and humid air. Explain. List all of your assumptions.

BIO Breathing You are breathing heavily while hiking up the mountain. To inhale, you expand your diaphragm and lungs. Explain, using your knowledge of gas pressure, why this mechanical movement leads to the air flowing into your nose or mouth. Support your reasoning with diagrams if necessary.

Oxygen tank for mountain climbing An oxygen container that one can use in the mountains has a $90-\min$ oxygen supply at a speed of 6 $\mathrm{L} / \mathrm{min}$ . Determine everything you can about the gas in the container. Make reasonable assumptions.

You are cooking dinner in the mountains. At 7000 feet, water boils at $92.3^{\circ} \mathrm{C}$ . Convert the boiling temperature to $^{\circ} \mathrm{F}$ and suggest two possible reasons why the boiling temperature is lower at this elevation than at sea level.

Your temperature, when taken orally, is $98.6^{\circ} \mathrm{F}$ . When taken under your arm, it's $36.6^{\circ} \mathrm{C}$ . Are these results consistent? Explain.

On top of Mount Everest, the temperature is $-19^{\circ} \mathrm{C}$ in July. Being a physicist, you determine by how many degrees Celsius one needs to change the air temperature to double the average kinetic energy of its molecules. Explain your reasoning.

Air consists of many different molecules, for example, $\mathrm{N}_{2}$ , $\mathrm{O}_{2}, \mathrm{H}_{2} \mathrm{O},$ and $\mathrm{CO}_{2}$ . Which molecules are the fastest on average? The slowest on average? Explain.

What is the average kinetic energy of a particle of air at standard conditions?

Air is a mixture of molecules of different types. Compare the rms speeds of the molecules of $\mathrm{N}_{2}, \mathrm{O}_{2},$ and $\mathrm{CO}_{2}$ at standard conditions. What assumptions did you make?

How many moles of air are in a regular $1-\mathrm{L}$ water bottle when you finish drinking the water? What assumptions did you make? How do these assumptions affect your result?

At approximately what temperature does the average random kinetic energy of a $\mathrm{N}_{2}$ molecule in an ideal gas equal the macroscopic translational kinetic energy of a copper atom in a penny that is dropped from the height of 1.0 $\mathrm{m} ?$

A molecule moving at speed $v_{1}$ collides head-on with a molecule of the same mass moving at speed $v_{2}$ . Compute the speeds of the molecules after the collision. What assumptions did you make? How does the answer to this problem explain why the mixing of hot and cold gases causes the cold gas to become warmer and the hot gas to become cooler?

Balloon flight For a balloon ride, the balloon must be inflated with helium to a volume of 1500 $\mathrm{m}^{3}$ at sea level. The balloon will rise to an altitude of about $12 \mathrm{km},$ where the temperature is about $-52^{\circ} \mathrm{C}$ and the pressure is about 20 $\mathrm{kPa}$ . How much helium should be put into the balloon? What assumptions did you make?

B10 Ears pop The middle ear has a volume of about 6.0 $\mathrm{cm}^{3}$ when at a pressure of $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}(1.0 \mathrm{atm}) .$ Determine the volume of that same air when the air pressure is $0.83 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ , as it is at an elevation of 1500 $\mathrm{m}$ above sea level (assume the air temperature remains constant). If the volume of the middle ear remains constant, some air will have to leave as the elevation increases. That is why ears “pop.”

Even the best vacuum pumps cannot lower the pressure in a container below $10^{-15}$ atm. How many molecules of air are left in each cubic centimeter in this "vacuum?" Assume that the temperature is 273 $\mathrm{K}$ .

Pressure in interstellar space The concentration of particles (assume neutral hydrogen atoms) in interstellar gas is 1 particle $/ \mathrm{cm}^{3},$ and the average temperature is about 10 $\mathrm{K}$ . What is the pressure of the interstellar gas? How does it compare to the best vacuum that can be achieved on Earth (see the previous problem)?

Describe experiments to determine if each of the three gas isoprocess laws works. The experiments should be ones that you could actually carry out.

The following data were collected for the temperature and volume of a gas. Can this gas be described by the ideal gas model? Explain how you know.

Describe a mechanical model of Stern’s molecular beam experiment.

Explain the microscopic mechanisms for the relation of macroscopic variables for an isothermal process, an isobaric process, and an isochoric process.

Scuba diving The pressure of the air in a diver's lungs when he is 20 $\mathrm{m}$ under the water surface is $3.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ , and the air occupies a volume of 4.8 $\mathrm{L}$ . How many moles of air should he exhale while moving to the surface, where the pressure is $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ ?

BIO EST Alveoli surface area Estimate the size of the surface area of a single alveolus in the lungs.

When surrounded by air at a pressure of $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ , a basketball has a radius of 0.12 $\mathrm{m}$ . Compare its volume at this condition with the volume that it would have if you take it 15 $\mathrm{m}$ below the water surface where the pressure is $2.5 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ . What assumptions did you make?

You have gas in a container with a movable piston. The walls of the container are thin enough so that its temperature stays the same as the temperature of the surrounding medium. You have baths of water of different temperatures, different objects that you can place on top of the piston, etc.
(a) Describe how you could make the gas undergo an isothermal process so that the pressure inside increases by $10 \%,$ then undergo an isobaric process so that the new volume decreases by 20$\%$ , and finally undergo an isochoric process so that the temperature increases by 15$\%$ . (b) Represent all processes in P-versus-T, V-versus-T, and $P$ -versus-V graphs. (c) What are the new pressure, volume, and temperature of the gas?

Bubbles While snorkeling, you see air bubbles leaving a crevice at the bottom of a reef. One of the bubbles has a radius of 0.060 m. As the bubble rises, the pressure inside it decreases by 50%. Now what is the bubble’s radius? What assumptions did you make to solve the problem?

Diving bell A cylindrical diving bell, open at the bottom and closed at the top, is 4 m tall. Scientists fill the bell with air at the pressure of $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ . The pressure increases by $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ for each 10 $\mathrm{m}$ that the bell is lowered below the surface of the water. If the bell is lowered 30 $\mathrm{m}$ below the ocean surface, how many meters of air space are left inside the bell? Why doesn’t water enter the entire bell as it goes under water? Draw several sketches for this problem.

Mount Everest (a) Determine the number of molecules per unit volume in the atmosphere at the top of Mount Everest. The pressure is $0.31 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ , and the temperature is $-30^{\circ} \mathrm{C}$ (b) Determine the number of molecules per unit volume at sea level, where the pressure is $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ and the temperature is $20^{\circ} \mathrm{C}$ .

EST Breathing on Mount Everest Using the information from Problem 44, estimate how frequently you need to breathe on top of Mount Everest to inhale the same amount of oxygen as you do at sea level. The pressure is about one-third the pressure at sea level.

Capping beer You would like to make homemade beer, but you are concerned about storing it. Your beer is capped into a bottle at a temperature of $27^{\circ} \mathrm{C}$ and a pressure of $1.2 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ . The cap will pop off if the pressure inside the bottle exceeds $1.5 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ . At what maximum temperature can you store the beer so the gas inside the bottle does not pop the cap? List the assumptions that you made.

Car tire With a tire gauge, you measure the pressure in a car tire as $2.1 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} .$ How can this be if you know that absolute pressure in the tire is three times higher than atmospheric? The tire looks okay. What's the deal?

Car tire dilemma Imagine a car tire that contains 5.1 moles of air when at a gauge pressure of $2.1 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ (the pressure above atmospheric pressure) and a temperature of $27^{\circ} \mathrm{C}$ . The temperature increases to $37^{\circ} \mathrm{C},$ the volume decreases to 0.8 times the original volume, and the gauge pressure decreases to $1.6 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ . Can these measurements be correct if the tire did not leak? If it did leak, then how many moles of air are left in the tire?

There is a limit to how much gas can pass through a pipeline, because the pipes can only tolerate so much pressure on the walls. To increase the amount of gas going through the pipe-line, engineers decide to cool the gas (to reduce its pressure). Suggest how much they should lower the temperature of the
gas if they want to increase the mass per unit time by 1.5 times.

Explain how you know that the volume of one mole of gas at standard conditions is 22.4 $\mathrm{L}$ .

At what pressure is the density of $-50^{\circ} \mathrm{C}$ nitrogen gas $\left(\mathrm{N}_{2}\right)$ equal to 0.10 times the density of water?

In the morning, the gauge pressure in your car tires is 35 psi. During the day, the air temperature increases from $20^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$ and the pressure increases to 36.5 $\mathrm{psi.}$ By how much did the volume of one of the tires increase? What assumptions did you make?

Equation Jeopardy 1 The equation below describes a process. Construct a word problem for a process that is consistent with the equations (there are many possibilities). Provide as much detailed information as possible about your proposed process.
$$\frac{1.2 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}}{293 \mathrm{K}}=\frac{2.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}}{T}$$

Equation Jeopardy 2 The equation below describes a process. Construct a word problem for a process that is consistent with the equations (there are many possibilities). Provide as much detailed information as possible about your proposed process.
$$\Delta n=\frac{\left(0.67 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\right)\left(0.60 \times 10^{-6} \mathrm{m}^{3}\right)}{(8.3 \mathrm{J} / \mathrm{mole} \cdot \mathrm{K})(303 \mathrm{K})}$$
$$\quad\quad -\frac{\left(1.00 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\right)\left(0.60 \times 10^{-6} \mathrm{m}^{3}\right)}{(8.3 \mathrm{J} / \mathrm{mole} \cdot \mathrm{K})(310 \mathrm{K})}$$

The $P$-versus-$T$ graph in Figure P9.55 describes a cyclic process comprising four hypothetical parts. (a) What happens to the pressure of the gas in each part? (b) What happens to the temperature of the gas in each part? (c) What happens to the volume of the gas in each part? (d) Explain each
part microscopically. (e) Use the information from (a)–(c) to represent the same parts in $P$-versus-$V$ and $V$ -versus-$T$graphs. [Hint: It helps to align the $P$-versus-$V$ graph beside the $P$-versus-$T$ graph using the same P values on the ordinate (vertical axes) and to place the $V$-versus-$T$graph below the $P$-versus-$T$ graph using the same $T$ values on the abscissa (horizontal axes). This helps keep the same scale for the variables.]

The $V$ -versus-$T$ graph in Figure P9.56 describes a cy clic process comprising four hypothetical parts. (a) What happens to the pressure of the gas in each parts? (b) What happens to the temperature of the gas in each parts? (c) What happens to the volume of the gas in each parts? (d) Explain each part microscopically. (e) Use the information from (a)–(c) to represent the same process in a $P$-versus-$T$ graph (below the $V$ -versus-$T$ graph) and a P-versus-V graph (beside the $P$-versus-$T$ graph). See the hint in the previous problem about the graph alignments.

EST Sun’s life expectancy (a) Estimate the average kinetic energy of the particles in the Sun. Assume that it is made of atomic hydrogen and that its average temperature is $100,000 \mathrm{K}$ . The mass of the Sun is $2 \times 10^{30} \mathrm{kg} .(\mathrm{b})$ For how long would the Sun shine using this energy if it radiates $4 \times 10^{26} \mathrm{W} / \mathrm{s} ?$ ls your answer reasonable? Explain.

The temperature of the Sun's atmosphere near the surface is about 6000 $\mathrm{K}$ , and the concentration of atoms is about $10^{15}$ particles/m $^{3}$ . What is the average pressure and density of its atmosphere? What assumptions did you make to solve the problem?

A gas that can be described by the ideal gas model is contained in a cylinder of volume $V$. The temperature of the gas is $T$. The particle mass is m, and the molar mass is $M$. Write an expression for the total thermal energy of the gas. Now, imagine that the exact same gas has been placed in a container of volume $2V$. What happens to its pressure? What happens to its temperature? What happens to its thermal energy?

BlO EST Breathing and metabolism We need about 0.7 $\mathrm{L}$ of oxygen per minute to maintain our resting metabolism and about 2 $\mathrm{L}$ when standing and walking. Estimate the number of breaths per minute for a person to satisfy this need when resting and when standing and walking. What assumptions did you make? Remember that oxygen is about 21$\%$ of the air.

In 1896, Lord Rayleigh showed that a mixture of two gases of different atomic masses could be separated by allowing some of it to diffuse through a porous membrane into an evacuated space. Rayleigh proposed that the molecules of lighter gas diffuse through the membrane faster, leaving the heavier gas behind in the original space. He described this "separation factor" as equal to $\sqrt{m_{2} / m_{1}},$ where $m_{1}$ is the molecular weight of a lighter gas and $m_{2}$ is the molecular weight of a heavier gas. Give a reason why the separation factor depends on the square root of the ratio of the molecular masses.

Equation Jeopardy 3 The three equations below describe a physical situation. Construct a word problem for a situation that is consistent with the equations (there are many possibilities). Provide as much detailed information as possible about the situation.
$$m=\left(1.3 \mathrm{kg} / \mathrm{m}^{3}\right)(3.0 \mathrm{m} \times 5.0 \mathrm{m} \times 2.0 \mathrm{m})$$
$$N=m /\left[\left(29 \times 10^{-3} \mathrm{kg}\right) /\left(6.0 \times 10^{23} \text { particles }\right)\right]$$
$$U_{\text { thermal }}=N(3 / 2)\left(1.38 \times 10^{-23} \mathrm{J} / \mathrm{K}\right)(273 \mathrm{K})$$

No $\mathrm{H}_{2}$ in Earth atmosphere Explain why Earth has almost no free hydrogen in its atmosphere.

No atmosphere on the Moon Why does the Moon have no atmosphere? Explain.

Different planet compositions Explain why planets closer to the Sun have low concentrations of light elements, but are relatively abundant in giant planets such as Jupiter, Uranus, and Saturn, which are far from the Sun.

EST Density of our galaxy Estimate the average density of particles in our galaxy, assuming that the most abundant element is atomic hydrogen. There are about $10^{9}$ stars in the galaxy and the size of the galaxy is about $10^{5}$ light-years. A light-year is the distance light travels in 1 year moving at a speed of $3 \times 10^{8} \mathrm{m} / \mathrm{s}$ . What do you need to assume about the stars in order to answer this question?

BIO Breathing Observe yourself breathing and count the number of times you inhale per second. During each breath you probably inhale about 0.50 L of air. How many oxygen molecules do you inhale if you are at sea level?

Car engine During a compression stroke of a cylinder in a diesel engine, the air pressure in the cylinder increases from $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ to $50 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ , and the temperature increases from $26^{\circ} \mathrm{C}$ to $517^{\circ} \mathrm{C}$ . Using this information, how would you convince your friends that knowledge about ideal gases can help explain how hot gases burned in the car engine affect the motion of the car?

EST How can the pressure of air in your house stay constant during the day if the temperature rises? Estimate the volume of your house and the number of moles of air that leave the house during the daytime. Assume that nighttime temperature and daytime temperature differ by about $10^{\circ} \mathrm{C}$ . List all other assumptions that are necessary to answer the question.

Tell-all problem Tell everything you can about the process described by the pressure-versus-volume graph shown in Figure P9.70.

Two massless, frictionless pistons are inside a horizontal tube opened at both ends. A 10 -cm-long thread connects the pistons. The cross-sectional area of the tube is 20 $\mathrm{cm}^{2} .$ The pressure and temperature of gas between the pistons and the outside air are the same and are equal to $P=1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ and $T=24^{\circ} \mathrm{C}$ . At what temperature will the thread break if it breaks when the tension reaches 30 $\mathrm{N}$ ?

A closed cylindrical container is divided into two parts by a light, movable, frictionless piston. The container's total length is 100 $\mathrm{cm}$ . Where is the piston located when one side is filled with nitrogen $\left(\mathrm{N}_{2}\right)$ and the other side with the same mass of hydrogen $\left(\mathrm{H}_{2}\right)$ at the same temperature?

Why is the wall tension in capillaries so small?
(a) There are so many capillaries.
(b) Their radii are so small.
(c) The outward pressure of the blood inside is so small.
(d) b and c
(e) a, b, and c

Which answer below is closest to the wall tension in a typical arteriole of radius 0.15 $\mathrm{mm}$ and 60 $\mathrm{mm}$ Hg blood pressure?
(a) 0.001 $\mathrm{N} / \mathrm{m} \quad$ (b) 0.01 $\mathrm{N} / \mathrm{m}$
(c) 0.1 $\mathrm{N} / \mathrm{m} \quad$ (d) 1 $\mathrm{N} / \mathrm{m}$
(e) 10 $\mathrm{N} / \mathrm{m}$

According to Laplace’s law, elevated blood pressure in an artery should cause the wall tension in the artery to do what?
(a) Increase $\quad$ (b) Remain unchanged
(c) Decrease $\quad$ (d) Impossible to decide

As a person ages, the fibers in arteries become less elastic and the wall tension increases. According to Laplace’s law, this will cause the blood pressure to do what?
(a) Increase $\quad$ (b) Remain unchanged
(c) Decrease $\quad$ (d) Impossible to decide

Aortic blowout occurs when part of the wall of the aorta becomes weakened. What does this cause?
(a) A bulge and increased radius of the aorta when the blood pressure inside increases
(b) An increased radius of the aorta, which causes increased tension in the wall
(c) An increased tension in the aorta, which causes the radius to increase
(d) a and b
(e) a, b, and c

What is closest to the volume of the Gamow bag?
(a) 50 $\mathrm{L} \quad$ (b) 100 $\mathrm{L}\quad$ (c) 200 $\mathrm{L} \quad$ (d) 500 $\mathrm{L} \quad$ (e) 1000 $\mathrm{L}$

What is closest to the temperature at the 6450-m elevation on the day described in the problem?
(a) 37 $\mathrm{K} \quad$ (b) 253 $\mathrm{K} \quad$ (c) 20 $\mathrm{K} \quad$ (d) $-20 \mathrm{K} \quad(\mathrm{e}) 273 \mathrm{K}$

What is closest to the number $n$ of gram-moles of air in the filled bag when at 4400 $\mathrm{m}$ ?
(a) 3 $\mathrm{g}\cdot$ moles $\quad$ (b) 10 $\mathrm{g} \cdot$ moles
(c) 13 $\mathrm{g}$ . moles $\quad$ (d) 110 $\mathrm{g} \cdot$ moles
(e) 170 $\mathrm{g} \cdot$ moles

What is closest to the number $n$ of gram-moles of air in the bag if at the 6450 -m pressure?
(a) 3 $\mathrm{g}\cdot$ moles $\quad$ (b) 10 $\mathrm{g} \cdot$ moles
(c) 13 $\mathrm{g}$ . moles $\quad$ (d) 110 $\mathrm{g} \cdot$ moles
(e) 170 $\mathrm{g} \cdot$ moles

Estimate the fraction of the air in the Gamow bag that an occupant would inhale in 1 h, assuming no replacement of the air.
(a) 0.01 $\quad$ (b) 0.1 $\quad$ (c) 0.3 $\quad$ (d) 0.5 $\quad$ (e) 1.0