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topic 1 Units, trigonometry, and Vectors
each of the following: (a) a mouse, (b) a pool cue, (c) a basketball
court, (d) an elephant, (e) a city block.
this information to estimate the mass of the following objects:
(a) a baseball, (b) your physics textbook, (c) a pickup truck.
month. (b) Estimate the number of human heartbeats in an
the diameter of an atom is about 10210 m.)
measured time intervals with their pulse. Why was this a
poor method of measuring time?
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.
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
other systems of units.
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).
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
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?
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.
is added to A
, 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?
that are equal in magnitude?
1.3 Dimensional Analysis
for one complete oscillation, is measured in time units
and is given by
T 5 2p
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.)
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 (2pft),
where A and f are constants. Find the dimensions of A. Hint:
A trigonometric function appearing in an equation must be
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.)
during an examination: (a) 12
mv2 5 12
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.
F 5 G
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 ?
It can be written in terms of the momentum p (Topic 6) and
mass m as
(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
width measured more accurately than the length. Find the
area, taking into account significant figures.
addition problem 21.4 1 15 1 17.17 1 4.003.
width 5.3 m. Find the area of the room retaining the proper
number of significant figures.
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.
3.788 3 109, (c) 2.46 3 1026, (d) 0.003 2
Express the speed of light to (a) three significant figures,
(b) five significant figures, and (c) seven significant figures.
6 0.1) m. Calculate (a) the area and (b) the perimeter of the
rectangle, and give the uncertainty in each value.
(a) the area and (b) the circumference of the circle, and
give the uncertainty in each value.
and 29 cm. Determine the volume of the box retaining the
proper number of significant figures in your answer.
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
to about 0.445 m. Convert the 2.00-m height of a basketball
forward to cubiti.
What is the area of this house in square meters?
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.
Find the speed of the turtle in centimeters per second. Note
that 1 furlong 5 220 yards and 1 fortnight 5 14 days.
How many cubic meters are there in 6.00 firkins?
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.
where the speed limit is 75.0 mi/h. Is the driver exceeding
the speed limit? Justify your answer.
gal). Express this efficiency in kilometers per liter (km/L).
(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.
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.
to miles per hour.
ceilings. What is the volume of the interior of the house in
cubic meters and in cubic centimeters?
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.
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 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
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
distance equal to the circumference of the Earth.
being during an average lifetime.
suffering from the common cold on any given day. (Answers
may vary. Remember that a person suffers from a cold for
about a week.)
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.
(a) Determine the volume of a cell. (b) Estimate the volume
of your body. (c) Estimate the number of cells in your body.
the number of revolutions the tire will make in its lifetime.
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
1.7 Coordinate systems
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.
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?
coordinates (5.0, 3.0) and (23.0, 4.0), where the units are
centimeters. Determine the distance between these points.
50.0) and (r, u) 5 (5.00 m, 250.0), respectively. What is the
distance between them?
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
length of the un known side, (b) the tangent of u, and (c) the
sine of f?
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?
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
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
to f, (c) cos u, (d) sin f, and (e) tan f.
to each other are 5.00 m and 7.00 m long. What is the length
of the third side of the triangle?
m is the opposite side?
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.
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?
has a magnitude of 8.00 units and makes an angle of
45.0 with the positive x axis. Vector B
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
and (b) the
vector difference A
has a magnitude of 29 units and points in the positive
y direction. When vector B
is added to A
, the resultant vector
points in the negative y direction with a magnitude of
14 units. Find the magnitude and direction of B
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
is 3.00 units in length and points along the positive
x axis. Vector B
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
and (b) A
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
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
1 1 F
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
the vector with magnitude 24.0 m and direction 56.0.
has components Ax 5 25.00 m and Ay 5 9.00 m.
Find (a) the magnitude and (b) the direction of the vector.
north and how far due east would she have to walk to arrive at
the same location?
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.
she coasts around one half of the circle, find (a) her distance
from the starting location and (b) the length of the path she
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?
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
of 40.0 units. Find the magnitude and direction of the
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.
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
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
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.
50 100 150 200
R S b S
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?
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?
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)
an area of 25.0 m2. What is the thickness of the fresh paint on
V 5 14/32pr 3. If the radius of sphere 2 is double the radius
of sphere 1, what is the ratio of (a) the areas, A2/A1 and
(b) the volumes, V2 /V1 ?
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?
(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.)
of Earths Moons could fit inside the Earth?
Estimate the worldwide number of sneezes happening in a
time interval approximately equal to one sneeze.
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
be zero? Explain.
be nonzero? Explain.
where the acceleration varies with time? (b) Can they be used
when the acceleration is zero?
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.
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?
of time ever be greater in magnitude than the average velocity
over a time interval containing that instant? (b) Can it ever
and acceleration when it reaches its maximum altitude?
(b) What is the acceleration of the ball just before it hits the
x1t 2 5 2t
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.
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.
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.
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.
2.1 Displacement, Velocity, and Acceleration
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
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.
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.
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
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?
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.
then stops for 15.0 minutes. He then continues north, traveling
What is his average velocity?
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.
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.
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
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.
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
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?
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?
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
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
maximum speed reached by the cells and (b) the distance
traveled during the acceleration.
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.
x 5 (5.0 m/s)t 1 (0.75 m/s3)t3
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).
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?
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
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?
(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!)
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?
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.
10 15 20
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
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.
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
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
15g for a short time.) Compare your answer with the free-fall
acceleration, 9.80 m/s2.
to a final velocity of 2.80 m/s. (a) Find the trucks original
speed. (b) Find its acceleration.
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 Dx.
(d) Substitute given values, obtaining that acceleration. (e)
Find the time it takes the boat to travel the given distance.
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?
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)?
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?
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?
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
for 15.0 s.
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?
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?
in 12.0 s. Find (a) the distance the car travels during this time
and (b) the constant acceleration of the car.
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?
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.
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
Mercedes Benz, which has continued at a constant speed?
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 v0 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.
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 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) 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?
of 8.00 m/s, from a height of 30.0 m. After what time interval
does it strike the ground?
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
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.
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
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?
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.
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?
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?
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.
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?
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.
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.
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?
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?
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
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.
x axis. Sketch plots of the objects position vs. time and velocity
a constant rate, and (c) its slowing down at a constant rate.
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?
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.
1.50 m/s after rising 2.00 m above its release point. Find the
balls initial speed.
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?
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
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?
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 v2 , v1.
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?
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
path where the velocity and acceleration vectors are (a) perpendicular
to each other? (b) Parallel to each other?
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.
acceleration: (a) a particle moving in a straight line with constant
speed and (b) a particle moving around a curve with constant
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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