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<< Test Bank For Understanding Business 11th Editionby by William Nickels | TEST BANK FOR THE ESSENCE OF ANTHROPOLOGY 4TH EDITION By HAVILAND >> |

**topic 1 ****Units, trigonometry, and Vectors **

**CONCEPTUAL QUESTIONS**

- Estimate the order of magnitude of the length, in meters, of

each of the following: (a) a mouse, (b) a pool cue, (c) a basketball

court, (d) an elephant, (e) a city block.

- What types of natural phenomena could serve as time

standards?

- Find the order of magnitude of your age in seconds.
- An object with a mass of 1 kg weighs approximately 2 lb. Use

this information to estimate the mass of the following objects:

(a) a baseball, (b) your physics textbook, (c) a pickup truck.

- (a) Estimate the number of times your heart beats in a

month. (b) Estimate the number of human heartbeats in an

average lifetime.

- Estimate the number of atoms in 1 cm3 of a solid. (Note that

the diameter of an atom is about 10210 m.)

- Lacking modern timepieces, early experimenters sometimes

measured time intervals with their pulse. Why was this a

poor method of measuring time?

- For an angle u measured from the positive
*x*axis, the values

of sin u and cos u are always (choose one): (a) greater than 11

(b) less than 21 (c) greater than 21 and less than 1 (d) greater

than or equal to 21 and less than or equal to 1 (e) less than or

equal to 21 or greater than or equal to 1.

- The left side of an equation has dimensions of length and the

right side has dimensions of length squared. Can the equation

be correct (choose one)? (a) Yes, because both sides involve

the dimension of length. (b) No, because the equation is

dimensionally inconsistent.

- List some advantages of the metric system of units over most

other systems of units.

- Estimate the time duration of each of the following in

the suggested units in parentheses: (a) a heartbeat (seconds),

(b) a football game (hours), (c) a summer (months), (d) a

movie (hours), (e) the blink of an eye (seconds).

- Suppose two quantities,
*A*and*B*, have different dimensions.

Determine which of the following arithmetic operations *could*

be physically meaningful. (a) *A *1 *B *(b) *B *2 *A *(c) *A *2 *B*

(d) *A*/*B *(e) *AB*

- Answer each question yes or no. Must two quantities have the

same dimensions (a) if you are adding them? (b) If you are

multiplying them? (c) If you are subtracting them? (d) If you

are dividing them? (e) If you are equating them?

- Two different measuring devices are used by students to measure

the length of a metal rod. Students using the first device

report its length as 0.5 m, while those using the second report

0.502 m. Can both answers be correct (choose one)? (a) Yes,

because their values are the same when both are rounded to

the same number of significant figures. (b) No, because they

report different values.

- If
**B**

S

is added to **A**

S

, under what conditions does the resultant

vector have a magnitude equal to *A *1 *B *? Under what conditions

is the resultant vector equal to zero?

- Under what circumstances would a vector have components

that are equal in magnitude?

**PROBLEMS**

**1.3 Dimensional Analysis**

- The period of a simple pendulum, defined as the time necessary

for one complete oscillation, is measured in time units

and is given by

*T *5 2p

,

*g*

where , is the length of the pendulum and *g *is the acceleration

due to gravity, in units of length divided by time squared.

Show that this equation is dimensionally consistent. (You

might want to check the formula using your keys at the end of

a string and a stopwatch.)

- (a) Suppose the displacement of an object is related to time

according to the expression *x *5 *Bt *2. What are the dimensions

of *B*? (b) A displacement is related to time as *x *5 *A *sin (2p*ft*),

where *A *and *f *are constants. Find the dimensions of *A. Hint:*

A trigonometric function appearing in an equation must be

dimensionless.

- A shape that covers an area
*A*and has a uniform height

*h *has a volume *V *5 *Ah. *(a) Show that *V *5 *Ah *is dimensionally

correct. (b) Show that the volumes of a cylinder and of a

rectangular box can be written in the form *V *5 *Ah, *identifying

*A *in each case. (Note that *A, *sometimes called the footprint

of the object, can have any shape and that the height can, in

general, be replaced by the average thickness of the object.)

- V Each of the following equations was given by a student

during an examination: (a) 12

*mv*2 5 12

*mv*0

2 1 !*mgh *(b) *v *5 *v *0

1 *at *2 (c) *ma *5 *v *2. Do a dimensional analysis of each equation

and explain why the equation cant be correct.

- Newtons law of universal gravitation is represented by

*F *5 *G*

*Mm*

*r *2

where *F *is the gravitational force, *M *and *m *are masses, and *r*

is a length. Force has the SI units kg ? m/s2. What are the SI

units of the proportionality constant *G *?

- Kinetic energy
*KE*(Topic 5) has dimensions kg ? m2/s2.

It can be written in terms of the momentum *p *(Topic 6) and

mass *m *as

*KE *5

*p*

2

2*m*

(a) Determine the proper units for momentum using dimensional

analysis. (b) Refer to Problem 5. Given the units of

force, write a simple equation relating a constant force *F*

exerted on an object, an interval of time *t *during which the

force is applied, and the resulting momentum of the object, *p.*

**1.4 Uncertainty in Measurement**

**and Significant Figures**

- A rectangular airstrip measures 32.30 m by 210 m, with the

width measured more accurately than the length. Find the

area, taking into account significant figures.

- Use the rules for significant figures to find the answer to the

addition problem 21.4 1 15 1 17.17 1 4.003.

- V A carpet is to be installed in a room of length 9.72 m and

width 5.3 m. Find the area of the room retaining the proper

number of significant figures.

- Use your calculator to determine (!8)3 to three significant

figures in two ways: (a) Find !8 to four significant figures;

then cube this number and round to three significant

figures. (b) Find !8 to three significant figures; then cube

this number and round to three significant figures. (c) Which

answer is more accurate? Explain.

- How many significant figures are there in (a) 78.9 6 0.2, (b)

3.788 3 109, (c) 2.46 3 1026, (d) 0.003 2

- The speed of light is now defined to be 2.997 924 58 3 108 m/s.

Express the speed of light to (a) three significant figures,

(b) five significant figures, and (c) seven significant figures.

- A rectangle has a length of (2.0 6 0.2) m and a width of (1.5

6 0.1) m. Calculate (a) the area and (b) the perimeter of the

rectangle, and give the uncertainty in each value.

- The radius of a circle is measured to be (10.5 6 0.2) m. Calculate

(a) the area and (b) the circumference of the circle, and

give the uncertainty in each value.

- The edges of a shoebox are measured to be 11.4 cm, 17.8 cm,

and 29 cm. Determine the volume of the box retaining the

proper number of significant figures in your answer.

- Carry out the following arithmetic operations: (a) the sum of

the measured values 756, 37.2, 0.83, and 2.5; (b) the product

0.003 2 3 356.3; (c) the product 5.620 3 p.

**1.5 Unit Conversions for Physical Quantities**

- The Roman cubitus is an ancient unit of measure equivalent

to about 0.445 m. Convert the 2.00-m height of a basketball

forward to cubiti.

- A house is advertised as having 1 420 square feet under roof.

What is the area of this house in square meters?

- A fathom is a unit of length, usually reserved for measuring

the depth of water. A fathom is approximately 6 ft in length.

Take the distance from Earth to the Moon to be 250 000 miles,

and use the given approximation to find the distance in fathoms.

- A small turtle moves at a speed of 186 furlongs per fortnight.

Find the speed of the turtle in centimeters per second. Note

that 1 furlong 5 220 yards and 1 fortnight 5 14 days.

- A firkin is an old British unit of volume equal to 9 gallons.

How many cubic meters are there in 6.00 firkins?

- Find the height or length of these natural wonders in kilometers,

meters, and centimeters: (a) The longest cave system in

the world is the Mammoth Cave system in Central Kentucky,

with a mapped length of 348 miles. (b) In the United States,

the waterfall with the greatest single drop is Ribbon Falls in

California, which drops 1 612 ft. (c) At 20 320 feet, Mount

McKinley in Alaska is Americas highest mountain. (d) The

deepest canyon in the United States is Kings Canyon in California,

with a depth of 8 200 ft.

- A car is traveling at a speed of 38.0 m/s on an interstate highway

where the speed limit is 75.0 mi/h. Is the driver exceeding

the speed limit? Justify your answer.

- A certain car has a fuel efficiency of 25.0 miles per gallon (mi/

gal). Express this efficiency in kilometers per liter (km/L).

- The diameter of a sphere is measured to be 5.36 in. Find

(a) the radius of the sphere in centimeters, (b) the surface

area of the sphere in square centimeters, and (c) the volume

of the sphere in cubic centimeters.

- V Suppose your hair grows at the rate of 1/32 inch

per day. Find the rate at which it grows in nanometers per second.

Because the distance between atoms in a molecule is on

the order of 0.1 nm, your answer suggests how rapidly atoms

are assembled in this protein synthesis.

- The speed of light is about 3.00 3 108 m/s. Convert this figure

to miles per hour.

- T A house is 50.0 ft long and 26 ft wide and has 8.0-ft-high

ceilings. What is the volume of the interior of the house in

cubic meters and in cubic centimeters?

- The amount of water in reservoirs is often measured in acre-ft.

One acre-ft is a volume that covers an area of one acre to a

depth of one foot. An acre is 43 560 ft2. Find the volume in SI

units of a reservoir containing 25.0 acre-ft of water.

- The base of a pyramid covers an area of 13.0 acres (1 acre 5

43 560 ft2) and has a height of 481 ft (Fig. P1.30). If the volume

of a pyramid is given by the expression *V *5 *bh*/3, where *b *is

the area of the base and *h *is the height, find the volume of this

pyramid in cubic meters.

- A quart container of ice cream is to be made in the form of

a cube. What should be the length of a side, in centimeters?

(Use the conversion 1 gallon 5 3.786 liter.)

**1.6 Estimates and order-of-Magnitude**

**Calculations**

*Note: *In developing answers to the problems in this section,

you should state your important assumptions, including

the numerical values assigned to parameters used in the

solution.

- Estimate the number of steps you would have to take to walk a

distance equal to the circumference of the Earth.

- V Estimate the number of breaths taken by a human

being during an average lifetime.

- Estimate the number of people in the world who are

suffering from the common cold on any given day. (Answers

may vary. Remember that a person suffers from a cold for

about a week.)

- The habitable part of Earths surface has been estimated to

cover 60 trillion square meters. Estimate the percent of this

area occupied by humans if Earths current population stood

packed together as people do in a crowded elevator.

- Treat a cell in a human as a sphere of radius 1.0 mm.

(a) Determine the volume of a cell. (b) Estimate the volume

of your body. (c) Estimate the number of cells in your body.

- An automobile tire is rated to last for 50 000 miles. Estimate

the number of revolutions the tire will make in its lifetime.

- A study from the National Institutes of Health states that

the human body contains trillions of microorganisms that make

up 1% to 3% of the bodys mass. Use this information to estimate

the average mass of the bodys approximately 100 trillion

microorganisms.

**1.7 Coordinate systems**

- T A point is located in a polar coordinate system by the

coordinates *r *5 2.5 m and u 5 35. Find the *x * and *y * coordinates

of this point, assuming that the two coordinate systems

have the same origin.

- A certain corner of a room is selected as the origin of a rectangular

coordinate system. If a fly is crawling on an adjacent wall at a

point having coordinates (2.0, 1.0), where the units are meters,

what is the distance of the fly from the corner of the room?

- Express the location of the fly in Problem 40 in polar

coordinates.

- V Two points in a rectangular coordinate system have the

coordinates (5.0, 3.0) and (23.0, 4.0), where the units are

centimeters. Determine the distance between these points.

- Two points are given in polar coordinates by (
*r,*u) 5 (2.00 m,

50.0) and (*r, *u) 5 (5.00 m, 250.0), respectively. What is the

distance between them?

- Given points (
*r*1, u1) and (*r*2, u2) in polar coordinates,

obtain a general formula for the distance between them. Simplify

it as much as possible using the identity cos2 u 1 sin2 u 5 1.

*Hint: *Write the expressions for the two points in Cartesian

coordinates and substitute into the usual distance formula.

**1.8 Trigonometry review**

- T For the triangle shown in Figure P1.45, what are (a) the

length of the un known side, (b) the tangent of u, and (c) the

sine of f?

6.00 m

9.00 m

_

_

**Figure P1.45**

- A ladder 9.00 m long leans against the side of a building. If

the ladder is inclined at an angle of 75.0 to the horizontal,

what is the horizontal distance from the bottom of the ladder

to the building?

- A high fountain of water is located at the center of a circular

pool as shown in Figure P1.47. Not wishing to get his feet

wet, a student walks around the pool and measures its circumference

to be 15.0 m. Next, the student stands at the edge of

the pool and uses a protractor to gauge the angle of elevation

at the bottom of the fountain to be 55.0. How high is the

fountain?

**Figure P1.47**

55.0_

- V A right triangle has a hypotenuse of length 3.00 m, and

one of its angles is 30.0. What are the lengths of (a) the side

opposite the 30.0 angle and (b) the side adjacent to the 30.0

angle?

- In Figure P1.49, find (a) the side opposite u, (b) the side adjacent

to f, (c) cos u, (d) sin f, and (e) tan f.

5.00

_

_

3.00

4.00

**Figure P1.49**

- In a certain right triangle, the two sides that are perpendicular

to each other are 5.00 m and 7.00 m long. What is the length

of the third side of the triangle?

- In Problem 50, what is the tangent of the angle for which 5.00

m is the opposite side?

- A woman measures the angle of elevation of a mountaintop

as 12.0. After walking 1.00 km closer to the mountain

on level ground, she finds the angle to be 14.0. Find the

mountains height, neglecting the height of the womans eyes

above the ground. *Hint: *Distances from the mountain (*x *and

*x *2 1 km) and the mountains height are unknown. Draw two

triangles, one for each of the womans locations, and equate

expressions for the mountains height. Use that expression to

find the first distance *x *from the mountain and substitute to

find the mountains height.

- A surveyor measures the distance across a straight river by the

following method: starting directly across from a tree on the

opposite bank, he walks *x *5 1.00 3 102 m along the riverbank

to establish a baseline. Then he sights across to the tree. The

angle from his baseline to the tree is u 5 35.0 (Fig. P1.53).

How wide is the river?

**1.9 Vectors**

- Vector
**A**

S

has a magnitude of 8.00 units and makes an angle of

45.0 with the positive *x * axis. Vector **B**

S

also has a magnitude

of 8.00 units and is directed along the negative *x * axis. Using

graphical methods, find (a) the vector sum **A**

S

1 **B**

S

and (b) the

vector difference **A**

S

2 **B**

S

.

- Vector
**A**

S

has a magnitude of 29 units and points in the positive

*y * direction. When vector **B**

S

is added to **A**

S

, the resultant vector

**A **S

1 **B**

S

points in the negative *y * direction with a magnitude of

14 units. Find the magnitude and direction of **B**

S

.

- An airplane flies 2.00 3 102 km due west from city A to

city B and then 3.00 3 102 km in the direction of 30.0 north

of west from city B to city C. (a) In straight-line distance,

how far is city C from city A? (b) Relative to city A, in what

direction is city C? (c) Why is the answer only approximately

correct?

- Vector
**A**

S

is 3.00 units in length and points along the positive

*x * axis. Vector **B**

S

is 4.00 units in length and points along the

negative *y * axis. Use graphical methods to find the magnitude

and direction of the vectors (a) **A**

S

1 **B**

S

and (b) **A**

S

2 **B**

S

.

- A force
**F**

S

1 of magnitude 6.00 units acts on an object at the

origin in a direction u 5 30.0 above the positive *x * axis

(Fig. P1.58). A second force **F**

S

2 of magnitude 5.00 units acts

on the object in the direction of the positive *y * axis. Find

graphically the magnitude and direction of the resultant force

**F **S

1 1 **F**

S

2.

**F**2

S

**F**1

S

u

**Figure P1.58**

- V A roller coaster moves 2.00 3 102 ft horizontally and then

rises 135 ft at an angle of 30.0 above the horizontal. Next, it

travels 135 ft at an angle of 40.0 below the horizontal. Use

graphical techniques to find the roller coasters displacement

from its starting point to the end of this movement.

**1.10 Components of a Vector**

- Calculate (a) the
*x*component and (b) the*y*component of

the vector with magnitude 24.0 m and direction 56.0.

- A vector
**A**

S

has components *Ax *5 25.00 m and *Ay *5 9.00 m.

Find (a) the magnitude and (b) the direction of the vector.

- A person walks 25.0 north of east for 3.10 km. How far due

north and how far due east would she have to walk to arrive at

the same location?

- V The magnitude of vector
**A**

S

is 35.0 units and points in

the direction 325 counterclockwise from the positive *x * axis.

Calculate the *x * and *y * components of this vector.

- A figure skater glides along a circular path of radius 5.00 m. If

she coasts around one half of the circle, find (a) her distance

from the starting location and (b) the length of the path she

skated.

- A girl delivering newspapers covers her route by traveling

3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east.

(a) What is her final position relative to her starting location?

(b) What is the length of the path she walked?

- A quarterback takes the ball from the line of scrimmage, runs

backwards for 10.0 yards, and then runs sideways parallel to

the line of scrimmage for 15.0 yards. At this point, he throws

a 50.0-yard forward pass straight downfield, perpendicular to

the line of scrimmage. How far is the football from its original

location?

- A vector has an
*x*component of 225.0 units and a*y*component

of 40.0 units. Find the magnitude and direction of the

vector.

- A map suggests that Atlanta is 730. miles in a direction 5.00

north of east from Dallas. The same map shows that Chicago

is 560. miles in a direction 21.0 west of north from Atlanta.

Figure P1.68 shows the location of these three cities. Modeling

the Earth as flat, use this information to find the displacement

from Dallas to Chicago.

**Figure P1.68**

Chicago

Dallas

Atlanta

21.0_

5.00_

- mi
- mi
- T The eye of a hurricane passes over Grand Bahama Island

in a direction 60.0 north of west with a speed of 41.0 km/h.

Three hours later the course of the hurricane suddenly shifts

due north, and its speed slows to 25.0 km/h. How far from

Grand Bahama is the hurricane 4.50 h after it passes over the

island?

- The helicopter view in Figure P1.70 shows two people pulling

on a stubborn mule. Find (a) the single force that is

equivalent to the two forces shown and (b) the force a third

person would have to exert on the mule to make the net

- A commuter airplane starts from an airport and takes the

route shown in Figure P1.71. The plane first flies to city *A*,

located 175 km away in a direction 30.0 north of east. Next,

it flies for 150. km 20.0 west of north, to city *B. *Finally, the

plane flies 190. km due west, to city *C. *Find the location of

city *C *relative to the location of the starting point.

**Figure P1.71**

*y *(km)

*A*

*x *(km)

50 100 150 200

150

200

*C*

30.0

110

20.0

250 *B*

50

100

*O*

**a**S

**c**S

**R **S **b **S

E

N

S

W

**Additional Problems**

- (a) Find a conversion factor to convert from miles per hour

to kilometers per hour. (b) For a while, federal law mandated

that the maximum highway speed would be 55 mi/h. Use the

conversion factor from part (a) to find the speed in kilometers

per hour. (c) The maximum highway speed has been raised to

65 mi/h in some places. In kilometers per hour, how much of

an increase is this over the 55-mi/h limit?

- The displacement of an object moving under uniform acceleration

is some function of time and the acceleration. Suppose

we write this displacement as *s *5 *ka mt n*, where *k *is a

dimensionless constant. Show by dimensional analysis that this

expression is satisfied if *m *5 1 and *n *5 2. Can the analysis give

the value of *k*?

- V Assume it takes 7.00 minutes to fill a 30.0-gal gasoline

tank. (a) Calculate the rate at which the tank is filled in gallons

per second. (b) Calculate the rate at which the tank is filled

in cubic meters per second. (c) Determine the time interval,

in hours, required to fill a 1.00-m3 volume at the same rate.

(1 U.S. gal 5 231 in.3)

- T One gallon of paint (volume 5 3.79 3 1023 m3) covers

an area of 25.0 m2. What is the thickness of the fresh paint on

the wall?

- A sphere of radius
*r*has surface area*A*5 4p*r*2 and volume

*V *5 14/32p*r *3. If the radius of sphere 2 is double the radius

of sphere 1, what is the ratio of (a) the areas, *A*2/*A*1 and

(b) the volumes, *V*2 /*V*1 ?

- T Assume there are 100 million passenger cars in the United

States and that the average fuel consumption is 20 mi/gal of

gasoline. If the average distance traveled by each car is 10 000

mi/yr, how much gasoline would be saved per year if average

fuel consumption could be increased to 25 mi/gal?

- V In 2015, the U.S. national debt was about $18 trillion.

(a) If payments were made at the rate of $1 000 per second,

how many years would it take to pay off the debt, assuming

that no interest were charged? (b) A dollar bill is about 15.5

cm long. If 18 trillion dollar bills were laid end to end around

the Earths equator, how many times would they encircle

the planet? Take the radius of the Earth at the equator to be

6 378 km. (*Note: *Before doing any of these calculations, try to

guess at the answers. You may be very surprised.)

- (a) How many Earths could fit inside the Sun? (b) How many

of Earths Moons could fit inside the Earth?

- An average person sneezes about three times per day.

Estimate the worldwide number of sneezes happening in a

time interval approximately equal to one sneeze.

- The nearest neutron star (a collapsed star made primarily of

neutrons) is about 3.00 3 1018 m away from Earth. Given that

the Milky Way galaxy (Fig. P1.81) is roughly a disk of diameter

, 1021 m and thickness , 1019 m, estimate the number of neutron

stars in the Milky Way to the nearest order of magnitude.

** **

** **

**topic 2 ****Motion in One Dimension **

** **

**CONCEPTUAL QUESTIONS**

- If the velocity of a particle is nonzero, can the particles acceleration

be zero? Explain.

- If the velocity of a particle is zero, can the particles acceleration

be nonzero? Explain.

- If a car is traveling eastward, can its acceleration be westward?

Explain.

- (a) Can the equations in Table 2.4 be used in a situation

where the acceleration varies with time? (b) Can they be used

when the acceleration is zero?

- Two cars are moving in the same direction in parallel lanes

along a highway. At some instant, car A is traveling faster than

car B. Does that mean the acceleration of A is greater than

that of B at that instant? (a) Yes. At any instant, a faster object

always has a larger acceleration. (b) No. Acceleration only

tells how an objects velocity is *changing *at some instant.

- Figure CQ2.6 shows strobe photographs taken of a disk moving

from left to right under different conditions. The time

interval between images is constant. Taking the direction to

the right to be positive, describe the motion of the disk in

each case. For which case is (a) the acceleration positive? (b)

the acceleration negative? (c) the velocity constant?

- (a) Can the instantaneous velocity of an object at an instant

of time ever be greater in magnitude than the average velocity

over a time interval containing that instant? (b) Can it ever

be less?

- A ball is thrown vertically upward. (a) What are its velocity

and acceleration when it reaches its maximum altitude?

(b) What is the acceleration of the ball just before it hits the

ground?

- An object moves along the
*x*axis, its position given by

*x*1*t *2 5 2*t*

- Which of the following cannot be obtained from

a graph of *x *vs. *t *? (a) The velocity at any instant (b) the acceleration

at any instant (c) the displacement during some time

interval (d) the average velocity during some time interval (e)

the speed of the particle at any instant.

- A ball is thrown straight up in the air. For which situation are

both the instantaneous velocity and the acceleration zero? (a)

On the way up (b) at the top of the flight path (c) on the way

down (d) halfway up and halfway down (e) none of these.

- A juggler throws a bowling pin straight up in the air. After

the pin leaves his hand and while it is in the air, which statement

is true? (a) The velocity of the pin is always in the

same direction as its acceleration. (b) The velocity of the

pin is never in the same direction as its acceleration. (c)

The acceleration of the pin is zero. (d) The velocity of the

pin is opposite its acceleration on the way up. (e) The velocity

of the pin is in the same direction as its acceleration on

the way up.

- A racing car starts from rest and reaches a final speed
*v*in a

time *t*. If the acceleration of the car is constant during this

time, which of the following statements must be true? (a) The

car travels a distance *vt*. (b) The average speed of the car is

*v*/2. (c) The acceleration of the car is *v*/*t*. (d) The velocity of

Figure CQ2.6 the car remains constant. (e) None of these.

** **

**PROBLEMS**

**2.1 Displacement, Velocity, and Acceleration**

- The speed of a nerve impulse in the human body is

about 100 m/s. If you accidentally stub your toe in the dark,

estimate the time it takes the nerve impulse to travel to your

brain.

- Light travels at a speed of about 3 3 108 m/s. (a) How many

miles does a pulse of light travel in a time interval of 0.1 s,

which is about the blink of an eye? (b) Compare this distance

to the diameter of Earth.

- A person travels by car from one city to another with different

constant speeds between pairs of cities. She drives for 30.0

min at 80.0 km/h, 12.0 min at 100 km/h, and 45.0 min at 40.0

km/h and spends 15.0 min eating lunch and buying gas. (a)

Determine the average speed for the trip. (b) Determine the

distance between the initial and final cities along the route.

- A football player runs from his own goal line to the opposing

teams goal line, returning to the fifty-yard line, all in 18.0 s.

Calculate (a) his average speed, and (b) the magnitude of his

average velocity.

- Two boats start together and race across a 60-km-wide lake and

back. Boat A goes across at 60 km/h and returns at 60 km/h.

Boat B goes across at 30 km/h, and its crew, realizing how far

behind it is getting, returns at 90 km/h. Turnaround times are

negligible, and the boat that completes the round trip first

wins. (a) Which boat wins and by how much? (Or is it a tie?)

(b) What is the average velocity

of the winning boat?

- A graph of position versus

time for a certain particle

moving along the *x * axis is

shown in Figure P2.6. Find

the average velocity in the

time intervals from (a) 0 to

2.00 s, (b) 0 to 4.00 s, (c) 2.00

s to 4.00 s, (d) 4.00 s to 7.00 s,

and (e) 0 to 8.00 s.

- V A motorist drives north for 35.0 minutes at 85.0 km/h and

then stops for 15.0 minutes. He then continues north, traveling

- km in 2.00 h. (a) What is his total displacement? (b)

What is his average velocity?

- A tennis player moves in a

straight-line path as shown

in Figure P2.8. Find her average

velocity in the time intervals

from (a) 0 to 1.0 s, (b) 0

to 4.0 s, (c) 1.0 s to 5.0 s, and

(d) 0 to 5.0 s.

- A jet plane has a takeoff speed of
*v*to 5 75 m/s and can move

along the runway at an average acceleration of 1.3 m/s2. If the

length of the runway is 2.5 km, will the plane be able to use

this runway safely? Defend your answer.

- Two cars travel in the same direction along a straight highway,

one at a constant speed of 55 mi/h and the other at 70 mi/h.

(a) Assuming they start at the same point, how much sooner

does the faster car arrive at a destination 10 mi away? (b) How

far must the faster car travel before it has a 15-min lead on the

slower car?

- The cheetah can reach a top speed of 114 km/h (71 mi/h).

While chasing its prey in a short sprint, a cheetah starts from

rest and runs 45 m in a straight line, reaching a final speed of 72

km/h. (a) Determine the cheetahs average acceleration during

the short sprint, and (b) find its displacement at *t *5 3.5 s.

- An athlete swims the length
*L*of a pool in a time*t*1 and

makes the return trip to the starting position in a time *t *2. If

she is swimming initially in the positive *x * direction, determine

her average velocities symbolically in (a) the first half

of the swim, (b) the second half of the swim, and (c) the

round trip. (d) What is her average speed for the round

trip?

- T A person takes a trip, driving with a constant speed of 89.5

km/h, except for a 22.0-min rest stop. If the persons average

speed is 77.8 km/h, (a) how much time is spent on the trip

and (b) how far does the person travel?

- A tortoise can run with a speed of 0.10 m/s, and a hare can

run 20 times as fast. In a race, they both start at the same time,

but the hare stops to rest for 2.0 minutes. The tortoise wins by

a shell (20 cm). (a) How long does the race take? (b) What is

the length of the race?

- To qualify for the finals in a racing event, a race car must

achieve an average speed of 250. km/h on a track with a total

length of 1.60 3 103. If a particular car covers the first half of

the track at an average speed of 230. km/h, what minimum

average speed must it have in the second half of the event to

qualify?

- A paper in the journal
*Current Biology*tells of some

jellyfish-like animals that attack their prey by launching stinging

cells in one of the animal kingdoms fastest movements.

High-speed photography showed the cells were accelerated

from rest for 700. ns at 5.30 3 107 m/s

- Calculate (a) the

maximum speed reached by the cells and (b) the distance

traveled during the acceleration.

- A graph of position versus time for a certain particle moving

along the *x * axis is shown in Figure P2.6. Find the instantaneous

velocity at the instants (a) *t *5 1.00 s, (b) *t *5 3.00 s,

(c) *t *5 4.50 s, and (d) *t *5 7.50 s.

- A race car moves such that its position fits the relationship

*x *5 (5.0 m/s)*t *1 (0.75 m/s3)*t*3

where *x *is measured in meters and *t *in seconds. (a) Plot a

graph of the cars position versus time. (b) Determine the

instantaneous velocity of the car at *t *5 4.0 s, using time intervals

of 0.40 s, 0.20 s, and 0.10 s. (c) Compare the average

velocity during the first 4.0 s with the results of part (b).

- Runner A is initially 4.0 mi west of a flagpole and is running

with a constant velocity of 6.0 mi/h due east. Runner B is initially

3.0 mi east of the flagpole and is running with a constant

velocity of 5.0 mi/h due west. How far are the runners from

the flagpole when they meet?

- V A particle starts from rest

and accelerates as shown in

Figure P2.20. Determine (a)

the particles speed at *t *5 10.0

s and at *t *5 20.0 s, and (b) the

distance traveled in the first

20.0 s.

- A 50.0-g Super Ball traveling

at 25.0 m/s bounces off

a brick wall and rebounds at

22.0 m/s. A high-speed camera records this event. If the ball is

in contact with the wall for 3.50 ms, what is the magnitude of

the average acceleration of the ball during this time interval?

- The average person passes out at an acceleration of 7
*g*

(that is, seven times the gravitational acceleration on Earth).

Suppose a car is designed to accelerate at this rate. How much

time would be required for the car to accelerate from rest to

60.0 miles per hour? (The car would need rocket boosters!)

- V A certain car is capable of accelerating at a rate of

0.60 m/s2. How long does it take for this car to go from a

speed of 55 mi/h to a speed of 60 mi/h?

- The velocity vs. time graph for an object moving along a

straight path is shown in Figure P2.24. **(i) **Find the average

acceleration of the object during the time intervals (a) 0 to 5.0

s, (b) 5.0 s to 15 s, and (c) 0 to 20 s. **(ii) **Find the instantaneous

acceleration at (a) 2.0 s, (b) 10 s, and (c) 18 s.

5

*t *(s)

6

8

2

0

4

4

2

8

6

*v *(m/s)

10 15 20

Figure P2.24

- A steam catapult launches a jet aircraft from the aircraft carrier

*John C. Stennis, *giving it a speed of 175 mi/h in 2.50 s. (a)

Find the average acceleration of the plane. (b) Assuming the

acceleration is constant, find the distance the plane moves.

**2.3 One-Dimensional Motion with Constant**

**Acceleration**

- Solve Example 2.5, Car Chase, by a graphical method. On

the same graph, plot position versus time for the car and the

trooper. From the intersection of the two curves, read the

time at which the trooper overtakes the car.

- T An object moving with uniform acceleration has a velocity

of 12.0 cm/s in the positive *x *direction when its *x * coordinate

is 3.00 cm. If its *x *coordinate 2.00 s later is 25.00 cm, what is

its acceleration?

- V In 1865 Jules Verne proposed sending men to the Moon

by firing a space capsule from a 220-m-long cannon with final

speed of 10.97 km/s. What would have been the unrealistically

large acceleration experienced by the space travelers

during their launch? (A human can stand an acceleration of

15*g *for a short time.) Compare your answer with the free-fall

acceleration, 9.80 m/s2.

- A truck covers 40.0 m in 8.50 s while uniformly slowing down

to a final velocity of 2.80 m/s. (a) Find the trucks original

speed. (b) Find its acceleration.

- A speedboat increases its speed uniformly from
*v i*5 20.0

m/s to *v f *5 30.0 m/s in a distance of 2.00 3 102 m. (a) Draw

a coordinate system for this situation and label the relevant

quantities, including vectors. (b) For the given information,

what single equation is most appropriate for finding the

acceleration? (c) Solve the equation selected in part (b) symbolically

for the boats acceleration in terms of *v f *, *vi*, and D*x.*

(d) Substitute given values, obtaining that acceleration. (e)

Find the time it takes the boat to travel the given distance.

- A Cessna aircraft has a liftoff speed of 120. km/h. (a) What

minimum constant acceleration does the aircraft require if it

is to be airborne after a takeoff run of 240. m? (b) How long

does it take the aircraft to become airborne?

- An object moves with constant acceleration 4.00 m/s2 and

over a time interval reaches a final velocity of 12.0 m/s. (a) If

its original velocity is 6.00 m/s, what is its displacement during

the time interval? (b) What is the distance it travels during this

interval? (c) If its original velocity is 26.00 m/s, what is its displacement

during this interval? (d) What is the total distance

it travels during the interval in part (c)?

- In a test run, a certain car accelerates uniformly from

zero to 24.0 m/s in 2.95 s. (a) What is the magnitude of the

cars acceleration? (b) How long does it take the car to change

its speed from 10.0 m/s to 20.0 m/s? (c) Will doubling the

time always double the change in speed? Why?

- A jet plane lands with a speed of 100 m/s and can accelerate

at a maximum rate of 25.00 m/s2 as it comes to rest. (a)

From the instant the plane touches the runway, what is the

minimum time needed before it can come to rest? (b) Can

this plane land on a small tropical island airport where the

runway is 0.800 km long?

- Speedy Sue, driving at 30.0 m/s, enters a one-lane tunnel.

She then observes a slow-moving van 155 m ahead traveling

at 5.00 m/s. Sue applies her brakes but can accelerate

only at 22.00 m/s2 because the road is wet. Will there be a

collision? State how you decide. If yes, determine how far into

the tunnel and at what time the collision occurs. If no, determine

the distance of closest approach between Sues car and

the van.

- A record of travel along a straight path is as follows:
- Start from rest with a constant acceleration of 2.77 m/s2

for 15.0 s.

- Maintain a constant velocity for the next 2.05 min.
- Apply a constant negative acceleration of 29.47 m/s2

for 4.39 s.

(a) What was the total displacement for the trip?

(b) What were the average speeds for legs 1, 2, and 3 of the

trip, as well as for the complete trip?

- A train is traveling down a straight track at 20 m/s when the

engineer applies the brakes, resulting in an acceleration of

21.0 m/s2 as long as the train is in motion. How far does the

train move during a 40-s time interval starting at the instant

the brakes are applied?

- A car accelerates uniformly from rest to a speed of 40.0 mi/h

in 12.0 s. Find (a) the distance the car travels during this time

and (b) the constant acceleration of the car.

- A car starts from rest and travels for 5.0 s with a uniform acceleration

of 11.5 m/s2. The driver then applies the brakes, causing

a uniform acceleration of 22.0 m/s2. If the brakes are

applied for 3.0 s, (a) how fast is the car going at the end of the

braking period, and (b) how far has the car gone?

- A car starts from rest and travels for
*t*1 seconds with a uniform

acceleration *a *1. The driver then applies the brakes, causing

a uniform acceleration *a *2. If the brakes are applied for *t *2

seconds, (a) how fast is the car going just before the beginning

of the braking period? (b) How far does the car go before the

driver begins to brake? (c) Using the answers to parts (a) and

(b) as the initial velocity and position for the motion of the

car during braking, what total distance does the car travel?

Answers are in terms of the variables *a *1, *a *2, *t *1, and *t *2.

- In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes

Benz are moving side by side down a straightaway at

71.5 m/s. The driver of the Thunderbird realizes that she must

make a pit stop, and she smoothly slows to a stop over a distance

of 250 m. She spends 5.00 s in the pit and then accelerates out,

reaching her previous speed of 71.5 m/s after a distance of 350

- At this point, how far has the Thunderbird fallen behind the

Mercedes Benz, which has continued at a constant speed?

- The kinematic equations can describe phenomena other than

motion through space and time. Suppose *x *represents a persons

bank account balance. The units of *x *would be dollars

($), and velocity *v *would give the rate at which the balance

changes (in units of, for example, $/month). Acceleration

would give the rate at which *v *changes. Suppose a person

begins with ten thousand dollars in the bank. Initial money

management leads to no net change in the account balance so

that *v*0 5 0. Unfortunately, management worsens over time so

that *a *5 22.5 3 102 $/month2. Assuming *a *is constant, find

the amount of time in months until the bank account is empty.

- T A hockey player is standing on his skates on a frozen pond

when an opposing player, moving with a uniform speed of 12

m/s, skates by with the puck. After 3.0 s, the first player makes

up his mind to chase his opponent. If he accelerates uniformly

at 4.0 m/s2, (a) how long does it take him to catch his opponent,

and (b) how far has he traveled in that time? (Assume

the player with the puck remains in motion at constant speed.)

- A train 4.00 3 102 m long is moving on a straight track with

a speed of 82.4 km/h. The engineer applies the brakes at a

crossing, and later the last car passes the crossing with a speed

of 16.4 km/h. Assuming constant acceleration, determine

how long the train blocked the crossing. Disregard the width

of the crossing.

**2.4 Freely Falling objects**

- A ball is thrown vertically upward with a speed of 25.0 m/s.

(a) How high does it rise? (b) How long does it take to reach

its highest point? (c) How long does the ball take to hit the

ground after it reaches its highest point? (d) What is its velocity

when it returns to the level from which it started?

- V A ball is thrown directly downward with an initial speed

of 8.00 m/s, from a height of 30.0 m. After what time interval

does it strike the ground?

- A certain freely falling object, released from rest, requires

1.50 s to travel the last 30.0 m before it hits the ground. (a)

Find the velocity of the object when it is 30.0 m above the

ground. (b) Find the total distance the object travels during

the fall.

- An attacker at the base of a castle wall 3.65 m high

throws a rock straight up with speed 7.40 m/s at a height

of 1.55 m above the ground. (a) Will the rock reach the

top of the wall? (b) If so, what is the rocks speed at the

top? If not, what initial speed must the rock have to reach

the top? (c) Find the change in the speed of a rock thrown

straight down from the top of the wall at an initial speed

of 7.40 m/s and moving between the same two points. (d)

Does the change in speed of the downward-moving rock

agree with the magnitude of the speed change of the rock

moving upward between the same elevations? Explain physically

why or why not.

- Traumatic brain injury such as concussion results when

the head undergoes a very large acceleration. Generally, an

acceleration less than 800 m/s2 lasting for any length of time

will not cause injury, whereas an acceleration greater than

1 000 m/s2 lasting for at least 1 ms will cause injury. Suppose

a small child rolls off a bed that is 0.40 m above the floor. If

the floor is hardwood, the childs head is brought to rest in

approximately 2.0 mm. If the floor is carpeted, this stopping

distance is increased to about 1.0 cm. Calculate the magnitude

and duration of the deceleration in both cases, to determine

the risk of injury. Assume the child remains horizontal

during the fall to the floor. Note that a more complicated fall

could result in a head velocity greater or less than the speed

you calculate.

- A small mailbag is released from a helicopter that is descending

steadily at 1.50 m/s. After 2.00 s, (a) what is the speed

of the mailbag, and (b) how far is it below the helicopter?

(c) What are your answers to parts (a) and (b) if the helicopter

is rising steadily at 1.50 m/s?

- A tennis player tosses a tennis ball straight up and then catches

it after 2.00 s at the same height as the point of release. (a)

What is the acceleration of the ball while it is in flight? (b)

What is the velocity of the ball when it reaches its maximum

height? Find (c) the initial velocity of the ball and (d) the

maximum height it reaches.

- A package is dropped from a helicopter that is descending

steadily at a speed *v *0. After *t *seconds have elapsed, (a)

what is the speed of the package in terms of *v *0, *g, *and *t *? (b)

What distance *d *is it from the helicopter in terms of *g *and *t *?

(c) What are the answers to parts (a) and (b) if the helicopter

is rising steadily at the same speed?

- A model rocket is launched straight upward with an initial

speed of 50.0 m/s. It accelerates with a constant upward

acceleration of 2.00 m/s2 until its engines stop at an altitude

of 150. m. (a) What can you say about the motion of the

rocket after its engines stop? (b) What is the maximum height

reached by the rocket? (c) How long after liftoff does the

rocket reach its maximum height? (d) How long is the rocket

in the air?

- V A baseball is hit so that it travels straight upward after

being struck by the bat. A fan observes that it takes 3.00 s for

the ball to reach its maximum height. Find (a) the balls initial

velocity and (b) the height it reaches.

**Additional Problems**

- A truck tractor pulls two trailers, one behind the other, at a

constant speed of 1.00 3 102 km/h. It takes 0.600 s for the big

rig to completely pass onto a bridge 4.00 3 102 m long. For

what duration of time is all or part of the trucktrailer combination

on the bridge?

- T Colonel John P. Stapp, USAF, participated in studying

whether a jet pilot could survive emergency ejection. On

March 19, 1954, he rode a rocket- propelled sled that moved

down a track at a speed of 632 mi/h (see Fig. P2.56). He and

the sled were safely brought to rest in 1.40 s. Determine in

SI units (a) the negative acceleration he experienced and (b)

the distance he traveled during this negative acceleration.

- A bullet is fired through a board 10.0 cm thick in such a way

that the bullets line of motion is perpendicular to the face of

the board. If the initial speed of the bullet is 4.00 3 102 m/s

and it emerges from the other side of the board with a speed

of 3.00 3 102 m/s, find (a) the acceleration of the bullet as it

passes through the board and (b) the total time the bullet is in

contact with the board.

- A speedboat moving at 30.0 m/s approaches a no-wake buoy

marker 1.00 3 102 m ahead. The pilot slows the boat with a

constant acceleration of 23.50 m/s2 by reducing the throttle.

(a) How long does it take the boat to reach the buoy? (b)

What is the velocity of the boat when it reaches the buoy?

- A student throws a set of keys vertically upward to his fraternity

brother, who is in a window 4.00 m above. The brothers

outstretched hand catches the keys 1.50 s later. (a) With what

initial velocity were the keys thrown? (b)? What was the velocity

of the keys just before they were caught?

- Mature salmon swim upstream, returning to spawn at

their birthplace. During the arduous trip they leap vertically

upward over waterfalls as high as 3.6 m. With what minimum

speed must a salmon launch itself into the air to clear a 3.6 m

waterfall?

- V An insect called the froghopper (
*Philaenus spumarius*)

has been called the best jumper in the animal kingdom. This

insect can accelerate at over 4.0 3 103 m/s2 during a displacement

of 2.0 mm as it straightens its specially equipped

jumping legs. (a) Assuming uniform acceleration, what is

the insects speed after it has accelerated through this short

distance? (b) How long does it take to reach that speed?

(c) How high could the insect jump if air resistance could

be ignored? Note that the actual height obtained is about

0.70 m, so air resistance is important here.

- An object is moving in the positive direction along the

*x * axis. Sketch plots of the objects position vs. time and velocity

- time if (a) its speed is constant, (b) its speeding up at

a constant rate, and (c) its slowing down at a constant rate.

- T A ball is thrown upward from the ground with an initial

speed of 25 m/s; at the same instant, another ball is dropped

from a building 15 m high. After how long will the balls be at

the same height?

- A player holds two baseballs a height
*h*above the ground.

He throws one ball vertically upward at speed *v *0 and the other

vertically downward at the same speed. Calculate (a) the

speed of each ball as it hits the ground and (b) the difference

between their times of flight.

- A ball thrown straight up into the air is found to be moving at

1.50 m/s after rising 2.00 m above its release point. Find the

balls initial speed.

- The thickest and strongest chamber in the human

heart is the left ventricle, responsible during systole for

pumping oxygenated blood through the aorta to rest of the

body. Assume aortic blood starts from rest and accelerates at

22.5 m/s2 to a peak speed of 1.05 m/s. (a) How far does the

blood travel during this acceleration? (b) How much time is

required for the blood to reach its peak speed?

- Emily challenges her husband, David, to catch a $1 bill

as follows. She holds the bill vertically as in Figure P2.67,

with the center of the bill

between Davids index finger

and thumb. David must catch

the bill after Emily releases

it without moving his hand

downward. If his reaction

time is 0.2 s, will he succeed?

Explain your reasoning. (This

challenge is a good trick you

might want to try with your

friends.)

- A mountain climber stands at the top of a 50.0 m cliff that

overhangs a calm pool of water. She throws two stones vertically

downward 1.00 s apart and observes that they cause a single

splash. The first stone had an initial velocity of 22.00 m/s. (a)

How long after release of the first stone did the two stones hit

the water? (b) What initial velocity must the second stone have

had, given that they hit the water simultaneously? (c) What was

the velocity of each stone at the instant it hit the water?

- One of Aesops fables tells of a race between a tortoise

and a hare. Suppose the overconfident hare takes a nap

and wakes up to find the tortoise a distance *d *ahead and a distance

*L *from the finish line. If the hare then begins running

with constant speed *v *1 and the tortoise continues crawling with

constant speed *v *2, it turns out that the tortoise wins the race if

the distance *L *is less than (*v *2 /(*v *1 2 *v *2))*d*. Obtain this result

by first writing expressions for the times taken by the hare and

the tortoise to finish the race, and then noticing that to win,

*t *tortoise , *t *hare. Assume *v*2 , *v*1.

- In Bosnia, the ultimate test of a young mans courage used

to be to jump off a 400 year old bridge (destroyed in 1993;

rebuilt in 2004) into the River Neretva, 23 m below the bridge.

(a) How long did the jump last? (b) How fast was the jumper

traveling upon impact with the river? (c) If the speed of sound

in air is 340 m/s, how long after the jumper took off did a

spectator on the bridge hear the splash?

- A stuntman sitting on a tree limb wishes to drop vertically

onto a horse galloping under the tree. The constant speed of

the horse is 10.0 m/s, and the man is initially 3.00 m above

the level of the saddle. (a) What must be the horizontal distance

between the saddle and the limb when the man makes

his move? (b) How long is he in the air?

** **

**topic 3 ****Motion in two Dimensions **

** **

**CONCEPTUAL QUESTIONS**

- As a projectile moves in its path, is there any point along the

path where the velocity and acceleration vectors are (a) perpendicular

to each other? (b) Parallel to each other?

- Construct motion diagrams showing the velocity and acceleration

of a projectile at several points along its path, assuming

(a) the projectile is launched horizontally and (b) the projectile

is launched at an angle u with the horizontal.

- Explain whether the following particles do or do not have an

acceleration: (a) a particle moving in a straight line with constant

speed and (b) a particle moving around a curve with constant

speed.

- A ball is projected horizontally from the top of a building. One

second later, another ball is projected horizontally from the same

point with the same velocity. (a) At what point in the motion will

the balls be closest to each other? (b) Will the first ball always be

traveling faster than the second? (c) What will be the time difference

between them when the balls hit the ground? (d) Can the

horizontal projection velocity of the second ball be changed so

that the balls arrive

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