Mechanics

A dog in an open field runs 12.0 m  east and then 28.0 m   in a direction  $50.0°$ west of north. In what direction and how far must the dog then run to end up 10.0 m  south of her original starting point?

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks$26.0 m$in a direction$60.0°$west of north. Jane walks$16.0 m$in a direction $30.0°$  south of west. They then stop and turn to face each other.

(a) What is the distance between them? 

(b) In what direction should Ricardo walk to go directly toward Jane?

In the methane molecule, ${\mathbf{CH}}_{\mathbf{4}}$, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the $\mathbf{C}\mathbf{-}\mathbf{H}$bonds is in the direction of $\hat{\mathbf{i}}\mathbf{+}\hat{\mathbf{j}}\mathbf{+}\hat{\mathbf{k}}$, an adjacent $\mathbf{C}\mathbf{-}\mathbf{H}$bond is in the $\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$direction. Calculate the angle between these two bonds.

Vectors $\stackrel{\rightharpoonup }{A}$and $\stackrel{\rightharpoonup }{B}$have scalar product -6.00 , and their vector product has magnitude +9.00 . What is the angle between these two vectors?

A cube is placed so that one corner is at the origin and three edges are along the x-, y-, and z-axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle between the edge along the z-axis (line ab) and the diagonal from the origin to the opposite corner (line ad), and (b) the angle between line ac (the diagonal of a face) and line ad.

Completed Pass. The football team at Enormous State University (ESU) uses vector displacements to record its plays, with the origin taken to be the position of the ball before the play starts. In a certain pass play, the receiver starts at $\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$ where the units are yards, data-custom-editor="chemistry" $\hat{\mathbf{i}}$ is to the right, and data-custom-editor="chemistry" $\hat{\mathbf{j}}$ is downfield. Subsequent displacements of the receiver are data-custom-editor="chemistry" $\mathbf{+}\mathbf{9}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}$ (he is in motion before the snap),  data-custom-editor="chemistry" $\mathbf{+}\mathbf{11}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$ (breaks downfield),  data-custom-editor="chemistry" $\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$ (zigs), and  data-custom-editor="chemistry" $\mathbf{+}\mathbf{12}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{14}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$ (zags). Meanwhile, the quarterback has dropped straight back to a position data-custom-editor="chemistry" $\mathbf{-}\mathbf{7}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$. How far and in which direction must the quarterback throw the ball? (Like the coach, you will be well advised to diagram the situation before solving this numerically.)

Navigating in the Big Dipper. All of the stars of the Big Dipper (part of the constellation Ursa Major) may appear to be the same distance from the earth, but in fact, they are very far from each other. Figure P1.91 shows the distances from the earth to each of these stars. The distances are given in light-years (ly), the distance that light travels in one year. One light-year equals $\mathbf{9}\mathbf{.}\mathbf{461}\mathbf{\times }{\mathbf{10}}^{\mathbf{15}}\mathbf{ }\mathbf{m}$  (a) Alkaid and Merak are data-custom-editor="chemistry" $\mathbf{25}\mathbf{.}{\mathbf{6}}^{\mathbf{o}}$ apart in the earth’s sky. In a diagram, show the relative positions of Alkaid, Merak, and our sun. Find the distance in light-years from Alkaid to Merak. (b) To an inhabitant of a planet orbiting Merak, how many degrees apart in the sky would Alkaid and our sun be?

CALCULATING LUNGS VALUE IN HUMANS. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $\mathbf{480}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}$ and that the average volume of a single alveolus is $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$. (The volume of a sphere is $\mathbf{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}$ and the area of a sphere is $\mathbf{A}\mathbf{=}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}$).

1.92: What is total volume of the gas-exchanging region of the lungs? (a) $\mathbf{2000}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$; (b) $\mathbf{2}\mathbf{ }{\mathbf{m}}^{\mathbf{3}}$; (c) $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{L}$; (d) $\mathbf{120}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{L}$.

CALCULATING LUNGS VALUE IN HUMANS. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $\mathbf{480}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}$ and that the average volume of a single alveolus is $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$. (The volume of a sphere is $\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}$ and the area of a sphere is data-custom-editor="chemistry" $\mathbf{A}\mathbf{=}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}$.) 

1.93 If we assume that alveoli are spherical, what is the diameter of a typical alveolus? (a)0.20 mm; (b) 2 mm; (c) 20 mm; (d) 200 mm.

CALCULATING LUNGS VALUE IN HUMANS. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $\mathbf{480}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}$ and that the average volume of a single alveolus is $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$. (The volume of a sphere is $\mathbf{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}$ and the area of a sphere is $\mathbf{A}\mathbf{=}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}$.) 

1.94 Individuals vary considerably in total lung volume.

Figure P1.94 shows the results of measuring the total lung volume and average alveolar volume of six individuals. From these data, what can you infer about the relationship among alveolar size, total lung volume, and number of alveoli per individual? As the total volume of the lungs increases, (a) the number and volume of individual alveoli increase; (b) the number of alveoli increases and the volume of individual alveoli decreases; (c) the volume of the individual alveoli remains constant and the number of alveoli increases; (d) both the number of alveoli and the volume of individual alveoli remain constant.

A small rock is thrown vertically upward with a speed of 22.0 m/s from the edge of the roof of a 30.0-m-tall building. The rock doesn’t hit the building on its way back down and lands on the street below. Ignore air resistance. (a) What is the speed of the rock just before it hits the street? (b) How much time elapses from when the rock is thrown until it hits the street?

Does the speedometer of a car measure speed or velocity? Explain.

The black dots at the top of Fig. represent a series of high-speed photographs of an insect flying in a straight line from left to right (in the positive x-direction). Which of the graphs in Fig. most plausibly depicts this insect’s motion?

Can an object with constant acceleration reverse its direction of travel? Can it reverse its direction twice? In both cases, explain your reasoning.

Under what conditions is average velocity equal to instantaneous velocity?

Is it possible for an object to be (a) slowing down while its acceleration is increasing in magnitude; (b) speeding up while its acceleration is decreasing? In both cases, explain your reasoning.

Under what conditions does the magnitude of the average velocity equal the average speed?

When a Dodge Viper is at Elwood’s Car Wash, a BMW Z3 is at Elm and Main. Later, when the Dodge reaches Elm and Main, the BMW reaches Elwood’s Car Wash. How are the cars’ average velocities between these two times related?

A driver in Massachusetts was sent to traffic court for speeding. The evidence against the driver was that a policewoman observed the driver’s car alongside a second car at a certain moment, and the policewoman had already clocked the second car going faster than the speed limit. The driver argued, “The second car was passing me. I was not speeding.” The judge ruled against the driver because, in the judge’s words, “If two cars were side by side, both of you were speeding.” If you were a lawyer representing the accused driver, how would you argue this case?

Three archers each fire four arrows at a target. Joe’s four arrows hit at points$10 cm$ above,$10 cm$ below,$10 cm$ to the left, and$10 cm$  to the right of the centre of the target. All four of Moe’s arrows hit within $1 \mathrm{cm}$  of a point $20 cm$  from the centre, and Flo’s four arrows hit within $1 cm$ of the centre. The contest judge says that one of the archers is precise but not accurate, another archer is accurate but not precise, and the third archer is both accurate and precise. Which description applies to which archer? Explain.

Can you have zero displacement and nonzero average velocity? Zero displacement and nonzero velocity? Illustrate your answers on an x-t graph.

Is the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ a unit vector? Is the vector $(3.0\hat{\mathrm{i}}-2.0\hat{\mathrm{j}})$ a unit vector? Justify your answers.

Can you have zero acceleration and nonzero velocity? Use a  graph to explain.

Can you have zero velocity and nonzero average acceleration? Zero velocity and nonzero acceleration? Use a  graph to explain, and give an example of such motion. 

An automobile is traveling west. Can it have a velocity toward the west and at the same time have an acceleration toward the east? Under what circumstances? 

For a spherical planet with mass $\mathit{M}$, volume  $\mathbf{V}$, and radius $\mathit{R}$, derive an expression for the acceleration due to gravity at the planet’s surface, $\mathit{g}$, in terms of the average density of the planet, $\mathrm{\rho }=\mathbf{M}/\mathbf{V}$, and the planet’s diameter, $\mathbf{D}=\mathbf{2}\mathbf{R}$. The table gives the values of  $\mathit{D}$ and $\mathit{g}$ for the eight major planets:

(a) Treat the planets as spheres. Your equation for $\mathit{g}$ as a function of  $\mathit{\rho }$ and $\mathit{D}$ shows that if the average density of the planets is constant, a graph of $\mathbf{g}$ versus  $\mathit{D}$ will be well represented by a straight line. Graph $\mathit{g}$ as a function of $\mathit{D}$  for the eight major planets. What does the graph tell you about the variation in average density? (b) Calculate the average density for each major planet. List the planets in order of decreasing density, and give the calculated average density of each. (c) The earth is not a uniform sphere and has greater density near its center. It is reasonable to assume this might be true for the other planets. Discuss the effect this nonuniformity has on your analysis. (d) If Saturn had the same average density as the earth, what would be the value of $\mathit{g}$ at Saturn’s surface?

A car travels in the +x-direction on a straight and level road. For the first 4.00 s of its motion, the average velocity of the car is $V_{av-x} = 6.25m/s$ . How far does the car travel in 4.00 s?

In an experiment, a shearwater (a seabird) was taken from its nest, flown 5150 km away, and released. The bird found its way back to its nest 13.5 days after release. If we place the origin at the nest and extend the +x-axis to the release point, what was the bird’s average velocity in  (a) for the return flight and (b) for the whole episode, from leaving the nest to returning?

You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 km/h (65 mi/h), and the trip takes 1 h and 50 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 km/h (43 mi/h). How much longer does the trip take?

Starting from a pillar, you run 200 m east (the +x-direction) at an average speed of 5.0 m/s and then run 280 m west at an average speed of 4.0 m/s to a post. Calculate (a) your average speed from pillar to post and (b) youraverage velocity from pillar to post.

Starting from the front door of a ranch house, you walk 60.0 m due east to a windmill, turn around, and then slowly walk 40.0 m west to a bench, where you sit and watch the sunrise. It takes you 28.0 s to walk from the house to the windmill and then 36.0 s to walk from the windmill to the bench. For the entire trip from the front door to the bench, what are your (a) average velocity and (b) average speed?

A Honda Civic travels in a straight line along a road. The car’s distance x from a stop sign is given as a function of time t by the equation$x\left(t\right)=\alpha t^2-\beta t^3$  where   $\alpha =1.50 m/s^2$and $\beta =0.0500m/s^2$ . Calculate the average velocity of the car for each time interval: (a) t = 0 to t = 2.00 s; (b) t = 0 to t = 4.00 s; (c) t = 2.00 s to t = 4.00 s.

A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by$x\left(t\right)=b{t}^2-c{t}^3$, where$b=2.40m/s^2$ and$c=0.120m/s^3$ . (a) Calculate the average velocity of the car for the time interval t = 0 to t = 10.0 s. (b) Calculate the instantaneous velocity of the car at t = 0, t = 5.0 s, and t = 10.0 s. (c) How long after starting from rest is the car again at rest?

A bird is flying due east. Its distance from a tall building is given by 

$x\left(t\right)=28.0m+\left(12.4m/s\right)t-\left(0.0450 m/s^3\right) t^3$.What is the instantaneous velocity of the bird when t = 8.00 s?

A physics professor leaves her house and walks along the sidewalk toward campus. After 5 min, it starts to rain, and she returns home. Her distance from her house as a function of time is shown in Fig. E2.10 At which of the labeled points is her velocity (a) zero? (b) constant and positive? (c) constant and negative? (d) increasing in magnitude? (e) decreasing in magnitude?

The official’s truck in Fig. 2.2 is at at and is at at ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{277}\mathbf{ }\mathbf{\text{ m}}$, at ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{16}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{ s}}$, and is at ${\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{19}\mathbf{ }\mathbf{\text{ m}}$ at ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{ s}}$. (a) Sketch two different possible x-t graphs for the motion of the truck. (b) Does the average velocity ${\mathit{v}}_{\mathbf{a}\mathbf{v}\mathbf{-}\mathbf{x}}$ during the time interval from ${\mathit{t}}_{\mathbf{1}}$ to ${\mathit{t}}_{\mathbf{2}}$ have the same value for both of your graphs? Why or why not?

Under constant acceleration the average velocity of a particle is half the sum of its initial and final velocities. Is this still true if the acceleration is not constant? Explain.

can you find out a vector quantity that has a magnitude of zero but components that are not zero? explain can the magnitude of vector be less than magnitude of any of its component? Explain

You throw a baseball straight up in the air so that it rises

to a maximum height much greater than your height. Is the magnitude

of the ball’s acceleration greater while it is being thrown or

after it leaves your hand? Explain.

Prove these statements: (a) As long as you can ignore the

effects of the air, if you throw anything vertically upward, it will

have the same speed when it returns to the release point as when it

was released. (b) The time of flight will be twice the time it takes

to get to its highest point.

A dripping water faucet steadily releases drops 1.0 s apart.

As these drops fall, does the distance between them increase, decrease,

or remain the same? Prove your answer.

If you know the initial position and initial velocity of a vehicle

and have a record of the acceleration at each instant, can you

compute the vehicle’s position after a certain time? If so, explain

how this might be done.

From the top of a tall building, you throw one ball straight up with speed ${\mathit{v}}_{\mathbf{0}}$ and one ball straight down with speed ${\mathit{v}}_{\mathbf{0}}$.

(a) Which ball has the greater speed when it reaches the ground?

(b) Which ball gets to the ground first? (c) Which ball has a greater displacement when it reaches the ground? (d) Which ball has traveled the greater distance when it hits the ground?

An object is thrown straight up into the air and feels no

air resistance. How can the object have an acceleration when it has

stopped moving at its highest point?

Neptunium. In the fall of 2002, scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about $60 \mathrm{kg}$. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium-237 has a density of $19.5 \mathrm{g}/\mathrm{cm}3$. What would be the radius of a sphere of this material that has a critical mass?

(a) The recommended daily allowance (RDA) of the trace metal magnesium is $410$ mg/day for males. Express this quantity in mg/day. (b) For adults, the RDA of the amino acid lysine is $12$ mg per kg of body weight. How many grams per day should a $75$-kg adult receive? (c) A typical multivitamin tablet can contain $2.0$ mg of vitamin B$2$ (riboflavin), and the RDA is $0.0030$ g/day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources? (d) The RDA for the trace element selenium is$0.000070$g/day. Express this dose in mg/day.

A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the car’s motion, the eastward component of the car’s velocity is given by ${\mathbf{v}}_{\mathbf{x}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{860}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}\mathbf{)}{\mathbf{t}}^{\mathbf{2}}$. What is the acceleration of the car when ${\mathbf{v}}_{\mathbf{x}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{/}\mathbf{s}$?

A turtle crawls along a straight line, which we will call the x-axis with the positive direction to the right. The equation for the turtle’s position as a function of time is $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{cm}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\hspace{0.33em}}\mathbf{cm}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{t}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0625}\mathbf{\hspace{0.33em}}\mathbf{cm}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{)}{\mathbf{t}}^{\mathbf{2}}$. (a) Find the turtle’s initial velocity, initial position, and initial acceleration. (b) At what time t is the velocity of the turtle zero? (c) How long after starting does it take the turtle to return to its starting point? (d) At what times t is the turtle a distance of from its starting point? What is the velocity (magnitude and direction) of the turtle at each of those times? (e) Sketch graphs of x versus t, $v_{\mathbf{x}}$ versus t, and ${\mathbf{a}}_{\mathbf{x}}$ versus t, for the time interval t = 0 to t = 40 s.

An astronaut has left the International Space Station to test a new space scooter.

 Her partner measures the following velocity changes, each taking place in a $\mathbf{10}\text{ - }\mathbf{s}$ interval. 

What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval?

 Assume that the positive direction is to the right. 

(a) At the beginning of the interval, the astronaut is moving toward the right along the x-axis at  $\mathbf{15}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$, and at the end of the interval she is moving toward the right at$5.0m/s$  . 

(b) At the beginning she is moving toward the left at  $5.0m/s$ , and at the end she is moving toward the left at  $15.0m/s$.

 (c) At the beginning she is moving toward the right at  , and at the end she is moving toward the left at  $15.0m/s$ .

Question: A car’s velocity as a function of time is given by${\mathbf{v}}_{\mathbf{x}}\left(\mathbf{t}\right)=\alpha +\beta {\mathbf{t}}^2$ , where$\alpha =\mathbf{3}.\mathbf{00} m/s$ and $\beta =\mathbf{0}.\mathbf{100}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{3}}$.  (a) Calculate the average acceleration for the time interval $\mathbf{t}=\mathbf{0} \mathbf{to} \mathbf{t}=\mathbf{5}.\mathbf{00} \mathbf{s}$. (b) Calculate the instantaneous acceleration for $\mathbf{t}=\mathbf{0} \mathbf{to} \mathbf{t}=\mathbf{5}.\mathbf{00} \mathbf{s}$. 

 (c) Draw ${\mathbf{v}}_{\mathbf{x}}\text{ - }\mathbf{t} \mathbf{and} {\mathbf{a}}_{\mathbf{x}}\text{ - }\mathbf{t}$graphs for the car’s motion between$\mathbf{t}=\mathbf{0} \mathbf{to} \mathbf{t}=\mathbf{5}.\mathbf{00} \mathbf{s}$.