time is 4.00 m west and 12.0 m above the center of the 30.0 cm diameter
29. An archer shoots an arrow at a 75.0 m distant target; the bull’s-eye of
nest. The owl is flying east at 3.50 m/s at an angle 30.0º below the
the target is at same height as the release height of the arrow. (a) At what horizontal when it accidentally drops the mouse. Is the owl lucky enough
angle must the arrow be released to hit the bull’s-eye if its initial speed is
to have the mouse hit the nest? To answer this question, calculate the
35.0 m/s? In this part of the problem, explicitly show how you follow the
horizontal position of the mouse when it has fallen 12.0 m.
steps involved in solving projectile motion problems. (b) There is a large
tree halfway between the archer and the target with an overhanging
42. Suppose a soccer player kicks the ball from a distance 30 m toward
horizontal branch 3.50 m above the release height of the arrow. Will the
the goal. Find the initial speed of the ball if it just passes over the goal,
arrow go over or under the branch?
2.4 m above the ground, given the initial direction to be 40º above the
30. A rugby player passes the ball 7.00 m across the field, where it is
horizontal.
caught at the same height as it left his hand. (a) At what angle was the
43. Can a goalkeeper at her/ his goal kick a soccer ball into the
ball thrown if its initial speed was 12.0 m/s, assuming that the smaller of
opponent’s goal without the ball touching the ground? The distance will
the two possible angles was used? (b) What other angle gives the same
be about 95 m. A goalkeeper can give the ball a speed of 30 m/s.
range, and why would it not be used? (c) How long did this pass take?
CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS 121
44. The free throw line in basketball is 4.57 m (15 ft) from the basket,
velocity of the wind? (b) If the bird turns around and flies with the wind,
which is 3.05 m (10 ft) above the floor. A player standing on the free
how long will he take to return 6.00 km? (c) Discuss how the wind affects
throw line throws the ball with an initial speed of 7.15 m/s, releasing it at
the total round-trip time compared to what it would be with no wind.
a height of 2.44 m (8 ft) above the floor. At what angle above the
54. Near the end of a marathon race, the first two runners are separated
horizontal must the ball be thrown to exactly hit the basket? Note that
by a distance of 45.0 m. The front runner has a velocity of 3.50 m/s, and
most players will use a large initial angle rather than a flat shot because it
the second a velocity of 4.20 m/s. (a) What is the velocity of the second
allows for a larger margin of error. Explicitly show how you follow the
runner relative to the first? (b) If the front runner is 250 m from the finish
steps involved in solving projectile motion problems.
line, who will win the race, assuming they run at constant velocity? (c)
45. In 2007, Michael Carter (U.S.) set a world record in the shot put with
What distance ahead will the winner be when she crosses the finish line?
a throw of 24.77 m. What was the initial speed of the shot if he released it
55. Verify that the coin dropped by the airline passenger in the Example
at a height of 2.10 m and threw it at an angle of 38.0º above the
3.8 travels 144 m horizontally while falling 1.50 m in the frame of
horizontal? (Although the maximum distance for a projectile on level
reference of the Earth.
ground is achieved at 45º when air resistance is neglected, the actual
56. A football quarterback is moving straight backward at a speed of 2.00
angle to achieve maximum range is smaller; thus, 38º will give a longer
m/s when he throws a pass to a player 18.0 m straight downfield. The
range than 45º in the shot put.)
ball is thrown at an angle of 25.0º relative to the ground and is caught at
the same height as it is released. What is the initial velocity of the ball
46. A basketball player is running at 5.00 m/s directly toward the basket relative to the quarterback ?
when he jumps into the air to dunk the ball. He maintains his horizontal
velocity. (a) What vertical velocity does he need to rise 0.750 m above
57. A ship sets sail from Rotterdam, The Netherlands, heading due north
the floor? (b) How far from the basket (measured in the horizontal
at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a
direction) must he start his jump to reach his maximum height at the
direction 40.0º north of east. What is the velocity of the ship relative to
same time as he reaches the basket?
the Earth?
47. A football player punts the ball at a 45.0º angle. Without an effect
58. A jet airplane flying from Darwin, Australia, has an air speed of 260
from the wind, the ball would travel 60.0 m horizontally. (a) What is the
m/s in a direction 5.0º south of west. It is in the jet stream, which is
initial speed of the ball? (b) When the ball is near its maximum height it
blowing at 35.0 m/s in a direction 15º south of east. What is the velocity
experiences a brief gust of wind that reduces its horizontal velocity by
of the airplane relative to the Earth? (b) Discuss whether your answers
1.50 m/s. What distance does the ball travel horizontally?
are consistent with your expectations for the effect of the wind on the
48. Prove that the trajectory of a projectile is parabolic, having the form
plane’s path.
y = ax + bx 2 . To obtain this expression, solve the equation x = v 0 xt 59. (a) In what direction would the ship in Exercise 3.57 have to travel in order to have a velocity straight north relative to the Earth, assuming its
for t and substitute it into the expression for y = v 0 yt – (1 / 2) gt 2
speed relative to the water remains 7.00 m/s ? (b) What would its speed
(These equations describe the x and y positions of a projectile that
be relative to the Earth?
starts at the origin.) You should obtain an equation of the form
60. (a) Another airplane is flying in a jet stream that is blowing at 45.0 m/s
y = ax + bx 2
in a direction 20º south of east (as in Exercise 3.58). Its direction of
where a and b are constants.
motion relative to the Earth is 45.0º south of west, while its direction of
v 2 sin 2θ
travel relative to the air is 5.00º south of west. What is the airplane’s
49. Derive R = 0
0
g
for the range of a projectile on level ground
speed relative to the air mass? (b) What is the airplane’s speed relative
by finding the time t at which y becomes zero and substituting this
to the Earth?
61. A sandal is dropped from the top of a 15.0-m-high mast on a ship
value of t into the expression for x − x 0 , noting that R = x − x 0
moving at 1.75 m/s due south. Calculate the velocity of the sandal when
50. Unreasonable Results (a) Find the maximum range of a super
it hits the deck of the ship: (a) relative to the ship and (b) relative to a
cannon that has a muzzle velocity of 4.0 km/s. (b) What is unreasonable
stationary observer on shore. (c) Discuss how the answers give a
about the range you found? (c) Is the premise unreasonable or is the
consistent result for the position at which the sandal hits the deck.
available equation inapplicable? Explain your answer. (d) If such a
62. The velocity of the wind relative to the water is crucial to sailboats.
muzzle velocity could be obtained, discuss the effects of air resistance,
Suppose a sailboat is in an ocean current that has a velocity of 2.20 m/s
thinning air with altitude, and the curvature of the Earth on the range of
in a direction 30.0º east of north relative to the Earth. It encounters a
the super cannon.
wind that has a velocity of 4.50 m/s in a direction of 50.0º south of west
51. Construct Your Own Problem Consider a ball tossed over a fence.
relative to the Earth. What is the velocity of the wind relative to the water?
Construct a problem in which you calculate the ball’s needed initial
velocity to just clear the fence. Among the things to determine are; the
63. The great astronomer Edwin Hubble discovered that all distant
height of the fence, the distance to the fence from the point of release of
galaxies are receding from our Milky Way Galaxy with velocities
the ball, and the height at which the ball is released. You should also
proportional to their distances. It appears to an observer on the Earth that
consider whether it is possible to choose the initial speed for the ball and
we are at the center of an expanding universe. Figure 3.64 illustrates this
just calculate the angle at which it is thrown. Also examine the possibility
for five galaxies lying along a straight line, with the Milky Way Galaxy at
of multiple solutions given the distances and heights you have chosen.
the center. Using the data from the figure, calculate the velocities: (a)
relative to galaxy 2 and (b) relative to galaxy 5. The results mean that
observers on all galaxies will see themselves at the center of the
expanding universe, and they would likely be aware of relative velocities,
52. Bryan Allen pedaled a human-powered aircraft across the English
concluding that it is not possible to locate the center of expansion with
Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1979. (a)
the given information.
He flew for 169 min at an average velocity of 3.53 m/s in a direction 45º
south of east. What was his total displacement? (b) Allen encountered a
headwind averaging 2.00 m/s almost precisely in the opposite direction of
his motion relative to the Earth. What was his average velocity relative to
the air? (c) What was his total displacement relative to the air mass?
53. A seagull flies at a velocity of 9.00 m/s straight into the wind. (a) If it
takes the bird 20.0 min to travel 6.00 km relative to the Earth, what is the
122 CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS
Figure 3.64 Five galaxies on a straight line, showing their distances and velocities
relative to the Milky Way (MW) Galaxy. The distances are in millions of light years
(Mly), where a light year is the distance light travels in one year. The velocities are
nearly proportional to the distances. The sizes of the galaxies are greatly
exaggerated; an average galaxy is about 0.1 Mly across.
64. (a) Use the distance and velocity data in Figure 3.64 to find the rate
of expansion as a function of distance.
(b) If you extrapolate back in time, how long ago would all of the galaxies
have been at approximately the same position? The two parts of this
problem give you some idea of how the Hubble constant for universal
expansion and the time back to the Big Bang are determined,
respectively.
65. An athlete crosses a 25-m-wide river by swimming perpendicular to
the water current at a speed of 0.5 m/s relative to the water. He reaches
the opposite side at a distance 40 m downstream from his starting point.
How fast is the water in the river flowing with respect to the ground?
What is the speed of the swimmer with respect to a friend at rest on the
ground?
66. A ship sailing in the Gulf Stream is heading 25.0º west of north at a
speed of 4.00 m/s relative to the water. Its velocity relative to the Earth is
4.80 m/s 5.00º west of north. What is the velocity of the Gulf Stream?
(The velocity obtained is typical for the Gulf Stream a few hundred
kilometers off the east coast of the United States.)
67. An ice hockey player is moving at 8.00 m/s when he hits the puck
toward the goal. The speed of the puck relative to the player is 29.0 m/s.
The line between the center of the goal and the player makes a 90.0º
angle relative to his path as shown in Figure 3.65. What angle must the
puck’s velocity make relative to the player (in his frame of reference) to
hit the center of the goal?
Figure 3.65 An ice hockey player moving across the rink must shoot backward to give
the puck a velocity toward the goal.
68. Unreasonable Results Suppose you wish to shoot supplies straight
up to astronauts in an orbit 36,000 km above the surface of the Earth. (a)
At what velocity must the supplies be launched? (b) What is
unreasonable about this velocity? (c) Is there a problem with the relative
velocity between the supplies and the astronauts when the supplies
reach their maximum height? (d) Is the premise unreasonable or is the
available equation inapplicable? Explain your answer.
69. Unreasonable Results A commercial airplane has an air speed of
280 m/s due east and flies with a strong tailwind. It travels 3000 km in a
direction 5º south of east in 1.50 h. (a) What was the velocity of the
plane relative to the ground? (b) Calculate the magnitude and direction of
the tailwind’s velocity. (c) What is unreasonable about both of these
velocities? (d) Which premise is unreasonable?
70. Construct Your Own Problem Consider an airplane headed for a
runway in a cross wind. Construct a problem in which you calculate the
angle the airplane must fly relative to the air mass in order to have a
velocity parallel to the runway. Among the things to consider are the
direction of the runway, the wind speed and direction (its velocity) and the
speed of the plane relative to the air mass. Also calculate the speed of
the airplane relative to the ground. Discuss any last minute maneuvers
the pilot might have to perform in order for the plane to land with its
wheels pointing straight down the runway.
CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION 123
4
DYNAMICS: FORCE AND NEWTON'S LAWS OF
MOTION
Figure 4.1 Newton’s laws of motion describe the motion of the dolphin’s path. (credit: Jin Jang)
Learning Objectives
4.1. Development of Force Concept
• Understand the definition of force.
4.2. Newton’s First Law of Motion: Inertia
• Define mass and inertia.
• Understand Newton's first law of motion.
4.3. Newton’s Second Law of Motion: Concept of a System
• Define net force, external force, and system.
• Understand Newton’s second law of motion.
• Apply Newton’s second law to determine the weight of an object.
4.4. Newton’s Third Law of Motion: Symmetry in Forces
• Understand Newton's third law of motion.
• Apply Newton's third law to define systems and solve problems of motion.
4.5. Normal, Tension, and Other Examples of Forces
• Define normal and tension forces.
• Apply Newton's laws of motion to solve problems involving a variety of forces.
• Use trigonometric identities to resolve weight into components.
4.6. Problem-Solving Strategies
• Understand and apply a problem-solving procedure to solve problems using Newton's laws of motion.
4.7. Further Applications of Newton’s Laws of Motion
• Apply problem-solving techniques to solve for quantities in more complex systems of forces.
• Integrate concepts from kinematics to solve problems using Newton's laws of motion.
4.8. Extended Topic: The Four Basic Forces—An Introduction
• Understand the four basic forces that underlie the processes in nature.
124 CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION
Introduction to Dynamics: Newton’s Laws of Motion
Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to achieve spectacular motion, such as that of a
dolphin jumping out of the water, or a pole vaulter, or the flight of a bird, or the orbit of a satellite. The study of motion is kinematics, but kinematics
only describes the way objects move—their velocity and their acceleration. Dynamics considers the forces that affect the motion of moving objects
and systems. Newton’s laws of motion are the foundation of dynamics. These laws provide an example of the breadth and simplicity of principles
under which nature functions. They are also universal laws in that they apply to similar situations on Earth as well as in space.
Issac Newton’s (1642–1727) laws of motion were just one part of the monumental work that has made him legendary. The development of Newton’s
laws marks the transition from the Renaissance into the modern era. This transition was characterized by a revolutionary change in the way people
thought about the physical universe. For many centuries natural philosophers had debated the nature of the universe based largely on certain rules of
logic with great weight given to the thoughts of earlier classical philosophers such as Aristotle (384–322 BC). Among the many great thinkers who
contributed to this change were Newton and Galileo.
Figure 4.2 Issac Newton’s monumental work, Philosophiae Naturalis Principia Mathematica, was published in 1687. It proposed scientific laws that are still used today to describe the motion of objects. (credit: Service commun de la documentation de l'Université de Strasbourg)
Galileo was instrumental in establishing observation as the absolute determinant of truth, rather than “logical” argument. Galileo’s use of the
telescope was his most notable achievement in demonstrating the importance of observation. He discovered moons orbiting Jupiter and made other
observations that were inconsistent with certain ancient ideas and religious dogma. For this reason, and because of the manner in which he dealt
with those in authority, Galileo was tried by the Inquisition and punished. He spent the final years of his life under a form of house arrest. Because
others before Galileo had also made discoveries by observing the nature of the universe, and because repeated observations verified those of
Galileo, his work could not be suppressed or denied. After his death, his work was verified by others, and his ideas were eventually accepted by the
church and scientific communities.
Galileo also contributed to the formation of what is now called Newton’s first law of motion. Newton made use of the work of his predecessors, which
enabled him to develop laws of motion, discover the law of gravity, invent calculus, and make great contributions to the theories of light and color. It is
amazing that many of these developments were made with Newton working alone, without the benefit of the usual interactions that take place among
scientists today.
It was not until the advent of modern physics early in the 20th century that it was discovered that Newton’s laws of motion produce a good
approximation to motion only when the objects are moving at speeds much, much less than the speed of light and when those objects are larger than
the size of most molecules (about 10−9 m in diameter). These constraints define the realm of classical mechanics, as discussed in Introduction to
the Nature of Science and Physics. At the beginning of the 20th century, Albert Einstein (1879–1955) developed the theory of relativity and, along
with many other scientists, developed quantum theory. This theory does not have the constraints present in classical physics. All of the situations we
consider in this chapter, and all those preceding the introduction of relativity in Special Relativity, are in the realm of classical physics.
Making Connections: Past and Present Philosophy
The importance of observation and the concept of cause and effect were not always so entrenched in human thinking. This realization was a part
of the evolution of modern physics from natural philosophy. The achievements of Galileo, Newton, Einstein, and others were key milestones in
the history of scientific thought. Most of the scientific theories that are described in this book descended from the work of these scientists.
4.1 Development of Force Concept
Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. Our intuitive
definition of force—that is, a push or a pull—is a good place to start. We know that a push or pull has both magnitude and direction (therefore, it is a
vector quantity) and can vary considerably in each regard. For example, a cannon exerts a strong force on a cannonball that is launched into the air.
In contrast, Earth exerts only a tiny downward pull on a flea. Our everyday experiences also give us a good idea of how multiple forces add. If two
people push in different directions on a third person, as illustrated in Figure 4.3, we might expect the total force to be in the direction shown. Since force is a vector, it adds just like other vectors, as illustrated in Figure 4.3(a) for two ice skaters. Forces, like other vectors, are represented by arrows and can be added using the familiar head-to-tail method or by trigonometric methods. These ideas were developed in Two-Dimensional
CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION 125
Figure 4.3 Part (a) shows an overhead view of two ice skaters pushing on a third. Forces are vectors and add like other vectors, so the total force on the third skater is in the direction shown. In part (b), we see a free-body diagram representing the forces acting on the third skater.
Figure 4.3(b) is our first example of a free-body diagram, which is a technique used to illustrate all the external forces acting on a body. The body is represented by a single isolated point (or free body), and only those forces acting on the body from the outside (external forces) are shown. (These
forces are the only ones shown, because only external forces acting on the body affect its motion. We can ignore any internal forces within the body.)
Free-body diagrams are very useful in analyzing forces acting on a system and are employed extensively in the study and application of Newton’s
laws of motion.
A more quantitative definition of force can be based on some standard force, just as distance is measured in units relative to a standard distance.
One possibility is to stretch a spring a certain fixed distance, as illustrated in Figure 4.4, and use the force it exerts to pull itself back to its relaxed
shape—called a