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Knight/Jones/Field Instructor Guide
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Chapter 6
Circular Motion, Orbits and Gravity
Recommended class days: 3
Background Information
This chapter deals with two different—but closely related—topics, circular motion and Newton's
Law of Gravity and orbital motion. Both of these topics require students to accept notions that are at
odds with their naive notions based on their everyday experience of the world. This is a
conceptually difficult chapter, but a chapter that is rich in applications from biology and other areas.
Circular motion requires a shift in thinking from the emphasis until now on linear motion and
motion that can be decomposed into independent linear motions along two axes. Not surprisingly,
this evokes new difficulties for students.
Arons (1990) cites an experiment in which a marble is rolled around the inside of an incomplete
circular hoop lying flat on a table. Students are asked to predict the marble’s trajectory after passing
point B. Some predict it will continue with perfect circular motion, reentering the hoop at A.
Others predict a circular tendency that “runs down,” so the marble curves to the left but gradually
straightens out. Either way, students seem to be thinking that the marble is subject to a “force of
motion” that has a memory of previous motions. Few predict the correct linear trajectory from B.
This experiment makes a great classroom demonstration to introduce the topic of circular motion.
Students are generally quite impressed that the ball takes a straight-line path after leaving the ring,
and this makes a great time to discuss other aspects of circular motion—such as the direction of the
force necessary to produce the motion.
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The Physics Education Research chapter cited the Van Heuvelen (1991a) study in which
students at the end of instruction were asked about the forces on a ball swinging in a vertical circle
on the end of a string. At the lowest point, 60% identified a “force of motion” tangent to the circle.
Further, if you look only at their vertical forces, 64% showed the tension force and the weight as
being of equal magnitude while only 36% recognized that the tension force would need to be larger
than the weight force. Swinging a ball in a circle in a horizontal circle makes another great
introductory demonstration. Students have just learned in Chapter 5 that tension forces act along the
string, so they can probably answer correctly that the force is toward the center of the circle . . . but
they really don't believe it. And then you can ask—and demonstrate—how the ball will move when
you let go of the string.
The term “centripetal force” causes difficulty for many students, and we avoid its use. They
consider it to be a new kind of force, like the gravitational force or the normal force, and they will
dutifully add a force vector labeled “centripetal force” to their free-body diagrams. We discuss the
centripetal acceleration, and then note that a force must exist to produce this acceleration. This
force could be one of the forces they are already familiar with, or a new one introduced in this
chapter—the long-range force of gravity.
Although satellite motion and celestial mechanics are somewhat peripheral to the text’s primary
focus on the physics of earthly objects, introducing gravity and orbits at this point does provide
another context for exploring the ideas of circular motion. Also, introducing gravity as a long-range
force is quite natural at this point after two chapters spent considering forces. And we must give
some historical information, as this is a topic that was central to the development of physics.
Some instructors like to introduce gravitational fields at this time, but we haven't had good luck
with this approach. Students have a difficult time with the concept of a field. Fields are very abstract
and intangible entities, and a major effort will be required for students to understand and use the
field model of interactions. Until these issues are dealt with, simply defining “the gravitational
field” is premature, devoid of any connection to physical phenomena or situations.
Student Learning Objectives
In covering the material of this chapter, students will
• Extend their understanding of the kinematics of circular motion beyond the basics presented in
Chapter 3.
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• Apply Newton’s laws in the context of circular motion, and solve a range of problems.
• Understand the circular orbits of satellites and planets as another case of circular motion.
• Develop an understanding of the long-range force of gravity.
Pedagogical Approach
The chapter starts with a discussion of circular motion. Discussing the kinematics and dynamics of
circular motion is a great way to apply the material of Chapter 1–5, but the concept of how circular
motion occurs as a consequence of center-directed forces is a very non-intuitive idea, and takes
some careful thought to introduce.
There have been many suggestions in recent years, especially in The American Journal of Physics,
that textbooks and instructors not use the term “centripetal force”—at least not at the beginning of
circular motion. As noted above, many students believe that this refers to a new and additional kind of
force acting on the object. We don’t label a tension force that causes linear acceleration “the linear
force,” so why should the same force get a special name if it happens to cause circular motion? The
term seems to hinder rather than help many students in their efforts to understand a difficult topic. As
noted, we use the term centripetal acceleration as the acceleration of an object moving in a circle, but
it does not use the term centripetal force. Instructors, if they intend to follow the text closely, need to
be careful not to let the term “slip out” during class discussion.
The idea of apparent weight returns in circular-motion examples of roller coasters, water in
buckets, jet planes doing loop-the-loops, and so on. These are more examples of reasoning than they
are of computation, but they are highly effective at forcing student attention on the principles of
Newtonian mechanics.
For the College Physics course, we think that there is no need to go beyond uniform circular
motion at this point. The interesting problems are ones for which the angular velocity is
approximately constant. Angular acceleration will be introduced in Chapter 7.
Finally, orbital motion is introduced in this chapter as an important application of circular motion.
This begins with the very counterintuitive notion orbital motion is nothing other than free fall—that a
satellite is “falling” around the earth but never getting any closer to the ground because the earth’s
surface curves away at the same rate. This concept needs a large variety of examples and explanations.
Newton’s law of gravity is something the students will have seen before, but it’s worthwhile
doing a careful introduction to remind students that this force is different than ones they have seen
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in Chapters 1–5. It’s a fundamental force of nature, and it involves a fundamental constant. Every
force they’ve encountered until now—tension, normal force, friction, etc.—has been situational.
The weight force, although given by the formula mg, depends on the local value of g. The friction
force was merely a model, not a law. The fundamental nature of the law of gravity is quite different,
and it is worth a short discussion to point this out.
Suggested Lecture Outlines
This material naturally breaks into three days: one day on basic concepts of circular motion, one
devoted to circular motion dynamics and examples, and one day on gravity and orbits. There should
be time for many interesting extensions. Some instructors may choose to get a “jump” on Chapter 7
on Day 3.
DAY 1: Basic Concepts. Until we get to Newton’s law of gravity, there is no “new” material in
this chapter. Circular motion was introduced in Chapter 3, and problems involving the dynamics of
circular motion are a natural extension of the material of Chapter 5, albeit with a different type of
acceleration.
Because much of this material is an extension of concepts students are already familiar with, we
suggest starting Day 1 with a series of interactive lecture demonstrations that give a review of the
material of previous chapters and illustrate the basic concepts of circular motion
Demonstration: Take a ball tightly fastened to a string about 1 m in length and swing it around
your head in a horizontal circle at a constant speed, fast enough that the string is nearly level. Ask
students to think about the motion, and ask them the following questions:
Clicker Question: We know that the ball is accelerating because
A. the speed is changing.
B. the direction is changing.
C. the speed and the direction are changing.
Clicker Question: What is the direction of the acceleration of the ball?
A. Tangent to the circle, in the direction of the ball’s motion
B. Toward the center of the circle
At this point, you should review the basic ideas of centripetal acceleration from Chapter 3, and then
continue with some questions about force:
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Clicker Question: What force is producing the centripetal acceleration of the ball?
A. Gravity
B. Air resistance
C. Normal force
D. Tension in the string
Clicker Question: What is the direction of the net force on the ball?
A. Tangent to the circle
B. Toward the center of the circle
C. There is no net force
If you lead the students to this point with the noted series of questions, they are likely to answer
these questions correctly. You may need to do a bit of discussion with the class, if there are doubts
or questions. Remind students that, clearly, the string plays a key role, and there is only one possible
direction that the string’s tension force can be directed.
Though students can answer these questions, some naive notions persist. It’s worthwhile to
follow with another demonstration—the demonstration alluded to at the start of the background
section.
It’s an easy matter to use a piece of flexible tubing and a marble on the level deck of an
overhead projector to set up a partial circle. A ball is given a quick flick and rolls around the circle
formed by the tube. But what happens when the ball reaches the break?
Clicker Question: When the ball reaches the break in the circle, which path will it follow?
This is a demonstration that is best done after students have had a chance to think about it, discuss it
with their neighbors, and “vote” on the answer. Once the students have had a chance to do this, do
the demonstration. We find that no matter how much they have thought about this experiment, it
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never fails to impress students; there are often audible noises of surprise. Students know that the ball
is supposed to go in a straight line where the tubing ends, but, somehow, they are still quite
surprised when it actually does! Their naive notions die hard.
With this as background and introduction, now is a good time to discuss the kinematics and
dynamics of circular motion in some detail, reviewing the material Chapters 1–5 along the way. Key
points to stress:
• In uniform circular motion, the speed is constant but the direction is constantly changing.
• The changing direction implies that there is an acceleration.
• The acceleration is directed toward the center of the circle—a centripetal acceleration.
• This acceleration must be produced by an identifiable force—tension in the string, the normal
force of the circular arc.
• There isn’t a new “centripetal force.” The centripetal acceleration is provided by one of the
forces we are already familiar with—or new ones we will learn about.
• When the force ceases to exist—as in the last demonstration—the object will simply continue in
a straight line.
Before continuing with a description of the dynamics of circular motion, students need practice
with the description and mathematics of circular motion. Students are encouraged to remember the
relationship between speed, radius, and period by recalling that speed  circumference/period. Thus
v  2 r / T.
Angular position  and angular velocity  are introduced at this time as the “natural”
mathematical language for circular motion. Even though they’ve seen radians before, nearly all
students need a review. This text is very explicit in including the radian units in expressions such as
  (2 rad)/T . At this point, you should review other units as well, and conversions between
them.
Example: The disk in a hard drive in a desktop computer rotates at 7200 rpm. The disk has a
diameter of 5.1 in (13 cm.) What is the angular speed of the disk?
This is a good example. It uses real data, and the result for the angular speed, 750 rad/s, is quite
high—necessary for rapid data writing and retrieval.
Once you have defined the necessary variables to describe circular motion, review the basic
equations for circular kinematics and dynamics:
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v  r
v2
a
  2r
r

r
r  mv 2
Fnet  ma  
, toward center of circle
 r

After this review, it’s time for some quantitative examples.
Example: The hard drive disk in the previous example rotates at 7200 rpm. The disk has a
diameter of 5.1 in (13 cm). What is the speed of a point 6.0 cm from the center axle? What is the
acceleration of this point on the disk?
Because the angular speed is so large, the acceleration is large as well—over 3000 g!
A classic example of circular dynamics presented in this chapter is that of the car turning a
corner, with the force toward the center provided by the friction between the tires and the road.
There are some worked-out examples in the text, but it’s worthwhile to do one in class as well, with
a full visual overview. This is another good place to practice the use of pictures as an aid to
understanding and solving problems. Here is another car turning a corner problem, with realistic
numbers.
Example: A level curve on a country road has a radius of 150 m. What is the maximum speed at
which this curve can be safely negotiated on a rainy day when the coefficient of friction between the
tires on a car and the road is 0.40?
The speed ends up being about 55 mph, though such a curve (a reasonably tight one) would
certainly have a sign suggesting a significantly lower speed, allowing a margin of safety.
DAY 2: Further development of circular motion dynamics. A lot of ground was covered
on Day 1. You might start Day 2 with a review, another circular dynamics problem, this one with
realistic numbers from the hammer throw.
Example: In the track and field event known as the hammer throw, an athlete spins a heavy mass
in a circle at the end of a chain. Once the mass gets moving at a good clip, the athlete lets go of the
chain. The mass flies off in a parabolic arc; the winner is the one who gets the maximum distance.
For male athletes, the “hammer” is a mass of 7.3 kg at the end of a 1.2 m chain. A world-class
thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal
circle centered on the handle he uses to hold the chain, what is the tension in the chain?
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Athletes competing in the hammer throw are quite strong; the forces involved are quite large, in
this case over 5000 N!
After an example with a horizontal circle, it’s time for an example with a vertical circle.
Demonstration: Most schools will have a “loop-the-loop” apparatus. A ball rolls down a ramp,
picking up speed, then rolls through a vertical loop. Ask your students to consider the force that
keeps the ball moving in the circle. What keeps the ball in the track even when it is upside down?
There’s no force that keeps the ball in the track—its inertia does the trick. But students are apt to
identify “centrifugal force” as the responsible agent. This is worth some class time to address.
After this demonstration, we suggest the following example, which addresses a situation with
which the students are certainly familiar. The following example uses realistic numbers for a real
country road.
Example: A car of mass 1500 kg goes over hill at a speed of 20 m/s. The shape of the hill is
approximately circular, with a radius of 60 m, as in the figure at right. When the car is at the highest
point of the hill,
a. What is the force of gravity on the car?
b. What is the normal force of the road on the car at this point?
At the top of the hill, the normal force is approximately 1/3 of the weight. If you were riding in
the car, you would “feel” light—the apparent weight is less than the actual weight. The concept of
apparent weight was first discussed in Chapter 5. At this point, a further discussion is in order.
This example leads naturally into a discussion of the maximum speed of walking, as discussed
in the chapter. When you walk, your body pivots over your forward foot, in an approximately
circular arc. The speed at which you can walk is determined by the length of your legs and by
gravity.
In the text, one example discussed with the concept of apparent weight is that of a bucket of
water swung over the head. Why does the water stay in the bucket? A related demonstration that is
more surprising to students is the “waiter’s tray.”
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Demonstration: Take a plate or tray and attach three strings or rope of approximately 1 m length.
Now, place a cup of water on the tray. If you are careful, you can swing this tray over your head
without spilling the water—which seems very surprising.
If you use a clear cup, students will be able to see that the level of the water stays the same relative
to the cup, a very dramatic illustration of the apparent weight. (In some countries, waiters actually
use trays similar to this; it’s interesting to watch someone move with such a tray. The stability is
quite remarkable.)
A related phenomenon that your students may have noticed is this: If you order a drink on an
airplane then set your cup on your tray table, when the airplane executes a turn, the cup does not
spill. The lift force on the plane is approximately perpendicular to the wings. To make a turn, the
plane must tip so that a component of this force points toward the center of the circular arc. The
net result is that the apparent weight points “down” toward the floor of the plane, not toward the
ground!
The chapter discusses the normal force—and the apparent weight—at different points for a roller
coaster car going through an inverted loop. It’s worthwhile to review this topic with a clicker
question.
Clicker Question: A roller coaster car goes through a vertical loop at a constant speed. For each
of the following quantities:
• centripetal acceleration
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• normal force
• apparent weight
make a rank ordering of positions A through E.
Ask students to think about the forces on the car at the top of the loop. It’s worthwhile to do a
quantitative example.
Example: A handful of professional skaters have taken a skateboard through an inverted loop in a
full pipe. For a typical pipe with diameter 14 ft, what is the minimum speed the skater must have at
the very top of the loop?
Ask your students to think about what happens if the speed is less than this.
At this point, it’s a good time to bring up Newton’s thought experiment concerning a ball
thrown from the top of a high mountain. If you can throw the ball fast enough, it will go into orbit.
“Down” is always toward the center of the earth, so, as the ball moves around the circle the
direction of the force changes, but the ball can stay at a constant height above the planet.
You may wish to repeat the calculation, in the chapter, of the orbital speed and period for an
earth-skimming orbit. Once you have thought about the orbit, come back to the idea of free fall
introduced in Chapter 5. An object in orbit is in free fall—so astronauts in the Space Shuttle or other
orbital craft will feel weightless.
For smaller celestial bodies, the orbital speed is quite a bit less, of course.
Example: Phobos is one of two small moons that orbit Mars. Phobos is a very small moon, and has
correspondingly small gravity—it varies, but a typical value is about 6 mm/s2. Phobos isn’t quite
round, but it has an average radius of about 11 km. What would be the orbital speed around Phobos,
assuming it was round with gravity and radius as noted?
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The orbital speed is only about 8 m/s; a modest softball toss would make it into orbit around
Phobos!
At this point, you might want to show a picture of Phobos. It’s clearly not round! Ask your
students to discuss why this might be true. Most students have not realized that it is gravity that
pulls planets and moons into their spherical shape—and that Phobos is too small to have sufficient
gravity to do this. Such a discussion is a good way to segue into the next topic, Newton’s law of
gravity.
DAY 3: Gravity and orbits, extensions. Newton’s law of gravity is the important concept to
introduce today. You might want to start with some history, perhaps relating the story of Newton
and the apple or talk about Newton’s calculations regarding the orbit of the moon—a nice way to
follow on what was done in the previous class. When you discuss gravity, there are some key points
to make:
• It’s a universal law.
• It’s an inverse-square law.
• It acts between any two objects in the universe.
“When you pick a flower, you move the most distant star in the universe.” —Paul Dirac
At this point, it’s worthwhile to do a couple of examples that illustrate why—though gravity acts
between all pairs of objects, we only notice its effects for massive, planet-sized objects.
Example: A typical bowling ball is spherical, weighs 16 pounds and has a diameter of 8.5 in.
Suppose two bowling balls are right next to each other in the rack. What is the gravitational force
between the two—magnitude and direction?
An example like this is important to show that the force of gravity is quite modest unless one (or
both) of the objects is quite large—as large as, perhaps, a planet or a moon. Follow up with a
calculation of the force of gravity on a person. Though this is done as an example in the chapter, it is
worth repeating, as it is an important point and there are some approximations involved that warrant
some discussion.
Example: What is the magnitude and direction of the force of gravity on a 60 kg person?
Of course, this force is just the person’s weight. You should make this connection, and share the
formula to compute g on other celestial bodies.
Clicker Question: A 60 kg person stands on each of the following planets. Rank order her weight
on the three bodies.
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After discussing gravity, move on to a discussion of orbits. This is a good way to connect with
the material of the early part of the chapter. An orbit is just a circular motion with the force provided
by gravity. We know how to compute gravitational forces and the dynamics of circular motion, so
we have all of the pieces that we need.
After presenting the basic equations, do an example or two.
Example: A spacecraft is orbiting the moon in an orbit very close to the surface—possible because
of the moon’s lack of atmosphere. What is the craft’s speed? The period of its orbit?
Example: Phobos is the closer of Mars’s two small moons, orbiting at 9400 km from the center of
Mars, a planet of mass 6.4  1023kg. What is Phobos’s orbital period? How does this compare to the
length of the Martian day, which is just shy of 25 hours?
Because its orbital period is less than the length of the Martian day, Phobos moves from west to
east across the Martian sky, while the sun and stars move east to west, as they do on earth.
After completing a basic discussion of gravity and orbits, there is time for some additional
topics. There are some very nice possibilities:
• The Coriolis effect. In the text, we don’t talk about “centrifugal force” and do not bring up the
idea of a pseudoforce. Instead, we talk about apparent weight. We also suggest not talking about
the “Coriolis force” but instead to discuss the Coriolis effect—something your students have
heard about and will likely wonder about.
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The main application of the Coriolis effect is to understanding motions in the atmosphere; air
masses in the northern hemisphere will follow paths that curve to the right. Because of this, air
masses that flow into the low pressure region at the center of a hurricane will develop a
counterclockise rotation. You might also address the common misconception about sinks and
toilets, which are far too small to exhibit any Coriolis-related rotation.
• If your students are primarily in the biological sciences, you might want to do some additional
discussion of the operation of centrifuges. Your students have the tools to do a calculation of the
time for small particles to settle out of solution using details of the drag force.
• You might want to do a bit more with orbital motion in the solar system, including a bit of
information on the closely related development of the understanding of these motions and all of
the physics of the book to this point.
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
suggestions for demonstrations, examples, questions, and additional topics that you could weave
into your class time.
Suggested Demonstrations
This is a good class for using videos (for instance, astronauts walking on the moon) and simulations
(there are many good simulations of orbital motion available) but there are some very good
demonstrations available. Some are noted above; here are some others:
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Tossing bottle with holes. Take a two-liter bottle and poke two ¼" holes on opposite sides near
the bottom. If you fill the bottle with water and then put on the cap, the water won’t leak, but as
soon as the cap is removed, two streams of water will come from the sides. Unless the bottle is
tossed in the air. . . . While it is in free fall, the weighlessness will keep the water from coming out.
A dramatic, fun—but a bit messy—demonstration.
Spinny dipper. This is a good one for showing apparent forces. A thin rectangular tank is
mounted on a bearing so that it can swivel.
The tank is filled halfway with water. Small boats floating on the water give it a nice additional
touch. When the tank is rotated, the water surface will form into a parabolic shape. “Down” for the
boats is perpendicular to the water surface, not toward the earth.
Sample Reading Quiz Questions
1. For uniform circular motion, the acceleration
A. is parallel to the velocity.
D.
is always due to gravity.
B. is directed toward the center of the circle.
E.
is always negative.
C. is larger for a larger orbit at the same speed.
2. When a car turns a corner on a level road, which force provides the necessary centripetal
acceleration?
A. Friction
D. Tension
B. Normal force
E. Air resistance
C. Gravity
3. Newton’s law of gravity describes the gravitational force between
A. the earth and the moon.
D. a person and the earth.
B. the earth and the sun.
E. all of the above.
C. the sun and the planets.
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Additional Student Response System (“Clicker”) Questions
1. A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher
orbit. Which of these quantities increases?
A. Speed
B. Angular speed
C. Period
D. Centripetal acceleration
E. Gravitational force of the earth
The following three questions concern the picture below, which shows a coin on a rotating
turntable.
2. At the time shown in the figure, which arrow gives the direction of the coin’s velocity?
3. At the time shown in the figure, which arrow gives the direction of the frictional force on the
coin?
4. At the instant shown, suppose the frictional force disappeared. In what direction would the
coin move?
Additional Examples
1. At Talladega, a NASCAR track, the turns have a 370 m radius and are banked at 33. At what
speed can a car go around this corner with no assistance from friction?
2. The Globe of Death, illustrated in the photo at the start of the chapter, is a spherical cage in
which motorcyclists ride in circular paths at high speeds. One outfit claims that riders achieve
a speed of 60 mph in a 16 ft diameter sphere. What would be the period for this motion? What
would be the apparent weight of a 60 kg rider at the bottom of the sphere? Given these two
pieces of information, does this high speed in this small sphere seem possible? (To help you
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decide, note that apparent weights for carnival rides are limited to 6 times a rider’s weight—
and then only for very short times.)
One Step Beyond: Locomotion
In the chapter, we discuss a model for understanding the maximum speed at which humans can
walk. The model gives pretty good results, and provides a reasonable explanation for many everyday
phenomena, such as why little children need to run to keep up with fast-walking adults. This is a
good topic to include for a typical College Physics class, with a high percentage of students in the
bio sciences. But you might want to take it a bit further, to look in more detail.
The formula for maximum walking speed predicts a maximum speed of 2.6 m/s for a typical
adult. But race walkers are capable of much higher speeds—over 4.0 m/s. How is this possible?
One way to walk faster is to bend the knees as you walk, keeping your center of mass level. If
you ask people to walk as fast as they can while keeping one foot on the ground at all times, this is a
common strategy they will use. But race walkers are required to keep their knees straight as they
pass under the body, so this won’t work for them. Instead, they use an unusual gait in which they
pivot the hips, with the same goal, to keep the body from rising and falling as much, effectively
increasing the radius of the circular arc through which the body goes. This lets them walk faster, but
if they go too fast, they run into trouble—their feet start leaving the ground, which will result in
disqualification.
Just because you can walk fast doesn’t mean that it is easier than running. As walking speeds
increase, so does the energy cost. At some point, it becomes more energetically favorable to run, and
that’s when most folks break into a run. You can walk at nearly 6 mph, but most folks will start
running at speeds that are less than this.
Four-legged animals have three different gaits that they can use; for a horse, these are a walk, a
trot and a gallop. Again, they will generally change gaits when it becomes energetically favorable to
do so. But large animals may avoid faster gaits, which result in higher skeletal stresses, and may
persist in a less efficient gait if it means less possibility for injury. Animals such as horses are
adapted to have long legs, which provide higher maximum speeds at all gaits, allowing them to
spend most of their time moving around at the slower, gentler gaits.
Walking on the moon is, as noted in the chapter, a slow affair. It would be even worse on
smaller bodies. On Phobos, the maximum walking speed would be a mere 6 cm/s! Actually, walking
would simply not be possible; big hops would be the way to go.
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