1. Describing Motion

2. Introduction
For example, Earth’s motion is a combination of a daily rotation about its axis and an annual revolution around the Sun.
Or, closer to home, the motion of a football can be treated as a combination of a vertical rise and fall, a horizontal movement, and a spinning about an axis.

3. Introduction
We therefore begin our discussion of motion by trying to describe and understand the simplest kinds of motion.
This will yield a conceptual framework within our world view from which even the most complicated motions, such as those associated with a hurricane or with a turbulent waterfall, can be understood.

4. Average Speed Imagine driving home from school.
For simplicity, assume that you can drive home in a straight line.
Normally, you might describe this trip in terms of the time it takes.
If pressed for a more detailed account, you would probably give the distance or the actual route taken, adding points of interest along the way.
For our purposes we need to develop a more precise description of motion.
First, we note that your position continually changes as you drive.
Second, we observe that it takes time to make the trip.
These two fundamental notions— space and time—are at the core of our concept of motion.
Furthermore, different positions along the trip can be matched with different times.

5. Average Speed
One relationship between space and time can be illustrated by answering the question, “How fast were you going?”
Actually, there are two ways to answer this question:
looking at the total trip, and
considering the moment-by-moment details of the trip.
For the total-trip description, we use the concept of an average speed, which is defined as the total distance traveled divided by the time it took to cover this distance.

6. Average Speed
We can write this relationship more efficiently by using symbols as abbreviations:
where is the average speed, d is the distance traveled, and t is the time taken for the trip.
A bar is often used over a symbol to indicate its average value.

7. Average Speed
This ratio of distance over time gives the average rate at which the car’s position changes.
Speed is a quantitative measure of how rapidly the change takes place.
The definition of average speed states a particular relationship between the concepts of space and time.
If any two of the three quantities are known, the third is determined.

8. Average Speed
To measure speed, we need a device for measuring distance, such as a ruler, and one for measuring time, such as a clock.
Most highways have mile markers along the side of the road so that maintenance and law enforcement officials can accurately find certain locations.
These mile markers and your wristwatch give you all the information you need to determine average speeds.

9. Average Speed
Assuming that we begin “thinking metric,” speeds have units such as meters per second (m/s) or kilometers per hour (km/h).
A person walks about 1½ meters per second, and a car traveling at 70 miles per hour (mph) is going approximately 113 kilometers per hour, or 113 km/h.

10. Images of Speed
As an example of measuring average speed, let’s determine the average speed of the puck in the figure.
The puck travels from a position near the 4-centimeter mark to one near the 76-centimeter mark, a total distance of 72 centimeters.
Because there are seven images, there are six intervals and the total time taken is six times the time between flashes—that is, 0.6 second.

11. Images of Speed Therefore, the average speed is
We can also determine the average speed of the puck between each pair of adjacent flashes.
Allowing for the uncertainties in reading the values of the positions of the puck, the average speed for each time interval is the same as the overall average.
Therefore, the puck was traveling at a constant speed of 120 centimeters per second.

12. Images of Speed
Suppose you live 40 miles from school and it takes you 2 hours to drive home.
Your average speed during the trip is
This means that, on the average, you travel a distance of 20 miles during each hour of travel.
This answer is read “20 miles per hour” and is often written as 20 miles/hour, or abbreviated as 20 mph.
It is important to include the units with your answer.

13. Images of Speed
A speed of “20” does not make any sense. It could be 20 miles per hour or 20 inches per year, very different average speeds.
Actually, you probably weren’t moving at 20 mph during much of your trip.
At times you may have been stopped at traffic lights;
at other times you may have traveled at 50 mph.
The use of average speed disregards the details of the trip.
Despite this, the concept of average speed is a useful notion.

14. On the Bus
Q: What is the average speed of an airplane that flies 3000 miles in 6 hours?
A: Using our definition for average speed, we have

15. Working It Out: Average Speed
If you know the average speed, you can determine other information about the motion.
For instance, you can obtain the time needed for a trip.
Suppose you plan to drive a distance of 60 miles with the cruise control set at 50 mph.
How long will the trip take?
Without consciously doing any calculation, you probably know that the answer is a little over 1 h.
How do you get a more precise answer?
You divide the distance traveled by the average speed.

16. Working It Out: Average Speed
For our example we obtain
You can also calculate how far you could drive if you traveled with a specified average speed for a specified time.
Suppose, for example, you plan to maintain an average speed of 50 mph on an upcoming trip.
How far can you travel if you drive an 8-h day?
Therefore, you would expect to drive 400 miles each day.

17. Instantaneous Speed
The notion of average speed is limited in most cases.
Even something as simple as your trip home from school is a much richer motion than our concept of average speed indicates.
For example, it doesn’t distinguish the parts of your trip when you were stopped waiting for a traffic light to change from those parts when you were exceeding the speed limit.
The simple question, “How fast were you going as you passed Third and Vine?” is not answered by knowing the average speed.

18. Instantaneous Speed
To answer the question, “How fast were you going at a specific point or at a specific time?” we need to consider a new concept known as the instantaneous speed.
This more complete description of motion tells us how fast you were traveling at any instant during your trip.
Because this is the function of your car’s speedometer, the idea is not new to you, although its precise definition may be new.

19. Instantaneous Speed
Actually, the definitions of average and instantaneous speeds are quite similar.
They differ only in the size of the time interval involved.
If we want to know how fast you are going at a given instant, we must study the motion during a very small time interval.
The instantaneous speed is equal to the average speed over a time interval that is very, very small.

20. Instantaneous Speed
As a first approximation in measuring the instantaneous speed, we could measure how far your car traveled during 1/10 of a second and calculate the average speed for this time interval.
With precise equipment we could determine the average speeds during time intervals of 1/100 of a second, 1/1000 of a second, or an even smaller interval.
How small an interval do we need?
For practical purposes, we need a time interval that is small enough that the average speed doesn’t change very much if we use an even smaller time interval.
It is the instantaneous speed rather than the average speed that plays an important role in the analysis of nearly all realistic motions.

21. Speed with Direction
We have made a lot of progress in attempting to accurately represent motion.
However, as we develop the rules for explaining (and thus predicting) the behavior of objects in the next chapter, we will need to go further.
Objects do more than speed up and slow down.
They can also change direction, sometimes keeping the same speed, but at other times changing both their speed and their direction.
Either the average speed or the instantaneous speed tells us how fast an object is moving, but neither tells us the direction of motion.

22. Speed with Direction
If we are discussing a vacation trip, direction doesn’t seem important; you obviously know in which direction you’re going.
However, we are trying to develop rules of motion for all situations, and the direction is as important as the speed.
You can get a sense for this by remembering situations in which there is an abrupt change in direction; for example, maybe a car you were riding in swerved sharply.
The squeal of the tires and your own body’s reaction are clues that new factors are involved when an object changes direction.

23. Speed with Direction
In the physics world view, we combine speed and direction into a single concept called velocity.
When we talk of an object’s instantaneous velocity, we give the instantaneous speed (for example, 15 mph) and just add the direction (north, to the left, or 30 degrees above the horizontal).
The speed is known as the magnitude of the velocity; it gives its size.
We use the symbol v to represent the magnitude of the instantaneous velocity.

24. Speed with Direction
Quantities that have both a size and a direction are called vectors.
Vectors do not obey the normal rules of arithmetic.
We will study the rules for combining vector quantities in Chapter 3.
For now, it is only important to realize that the direction of the motion can be as important as the speed.

25. Speed with Direction
There is another important difference between average speed and average velocity besides direction.
The average speed is defined as the distance traveled divided by the time taken, whereas the average velocity is defined as the displacement divided by the time taken.
Displacement is a vector quantity;
its magnitude is the straight-line distance between the initial and final locations of the object, and
its direction is from the initial location to the final location.

26. On the Bus
Q: A car travels due north a distance of 50 kilometers, turns around, and returns to the starting place along the same route. What distance did the car travel, and what was its displacement?
A: The car travels a distance of 100 kilometers, but it has a displacement of zero because it returns to its starting place.

27. Speed with Direction
The magnitude of the average velocity of an object is the change in position divided by the time taken to make the change:
We have used x to represent the position of the object.
The symbol Dx is called “delta ex.”
The delta symbol D is used to represent a change in a quantity.
Thus, Dx represents the change in position—the displacement—and must not be thought of as the product of D and x.

28. Speed with Direction
To calculate the displacement, we subtract the position of the object at the beginning of the time interval from its position at the end.
For example, if a car travels from milepost 120 to milepost 180, the displacement is
180 miles miles = 60 miles.
Notice that we have also written the time taken as Dt to indicate that it is an interval of time rather than an instant of time.

29. Acceleration
Because the velocities of many things are not constant, we need a way to describe how velocity changes.
We now define a new concept, called acceleration, which describes the rate at which velocity changes.
The magnitude of the average acceleration of an object is the change in its velocity divided by the time it takes to make that change:
As we did with speed, we can speak of either the average acceleration or the instantaneous acceleration, depending on the size of the time interval.

30. Acceleration
The units of acceleration are a bit more complicated than those of speed and velocity.
Remember that the units of velocity are distance divided by time:
for example, miles per hour or meters per second.
Because acceleration is the change in velocity divided by the time interval, its units are (distance per time) per time
for example, (kilometers per hour) per second or (meters per second) per second (abbreviated m/s/s or m/s2).

31. Acceleration The concept of acceleration is probably familiar to you.
We talk about one car having “better acceleration” than another.
This usually means that it can obtain a high speed in a shorter time.
For instance, a Dodge Grand Caravan can accelerate from 0 to 60 miles per hour in 11.3 seconds;
a Ford Taurus requires 8.7 seconds;
a Chevrolet Corvette requires only 4.8 seconds.

32. On the Bus Q: Which car has the largest average acceleration?
A: The Corvette has the largest average acceleration because it reaches 60 mph in the shortest time interval.

33. Acceleration
Another way of becoming more familiar with acceleration is by experiencing it.
For example, when an elevator begins to move up (or down) rapidly, the sensation you get in your stomach is due to the elevator (and you) quickly changing speed.
Astronauts feel this when the space shuttle blasts off from its launch pad.
Exciting examples of the same effect can be achieved on a roller coaster.
In fact, amusement parks can be thought of as places where people pay money to experience the effects of acceleration.

34. Acceleration
In contrast, you don’t feel motion when you’re traveling in a straight line at a constant speed—that is, motion with zero acceleration.
The motion you do feel when riding in a car on a straight highway is due to small vibrations of the car.
These vibrations are tiny changes in direction or small accelerations of the car caused by bumps in the road.
If you are standing on the sidewalk watching a moving object, how can you tell whether the object is accelerating?
One way is to take a strobe photograph of its motion.

35. Acceleration Which of the two corresponds to the car accelerating?
If you answered the second car, you have a qualitative understanding of acceleration.
First car travels the same distance during each time interval and therefore is traveling at a constant speed.
Second car travels farther during each successive time interval; it is accelerating.

36. Acceleration
Even if the car were slowing down, it would be accelerating.
We don’t usually use the word deceleration in physics because the word acceleration includes slowing down as well as speeding up.
In this case the distance traveled during successive time intervals would be shorter.
Acceleration refers to any change in speed or direction—that is, to any change in velocity.
Acceleration is a vector quantity.
When the acceleration is in the same direction as the velocity, the speed of the object is increasing.
When the acceleration and the velocity point in opposite directions, the object is slowing down.

37. Acceleration
The idea of an acceleration having a direction might seem a little abstract and, perhaps, unnecessary.
A car’s velocity obviously has a direction and probably seems easier to comprehend.
However, as we continue our study of acceleration, we will see many examples in which the direction of the acceleration has physical consequences.
The discussion of accelerations due only to a change in the direction of the velocity appears in Chapter 4.

38. On the Bus
Q: You see two cars side by side as they exit a tunnel. The red car has a speed of 40 meters per second and an acceleration of 20 (meters per second) per second. The blue car has a speed of 20 meters per second and an acceleration of 40 (meters per second) per second. At the instant they leave the tunnel, which car is passing the other?
A: The car with the larger instantaneous speed will travel farther down the road in the next small interval of time. Therefore, the red car is passing the blue car as they exit the tunnel. The blue car will have the greater change on its speedometer in the next small interval of time and will eventually overtake and pass the red car.

39. On the Bus
Q: What is an example from everyday life of something that is slowing with an acceleration vector pointing upward?
A: If this “something” is slowing, its velocity vector must be pointing in the opposite direction of its acceleration vector, or downward. Therefore, we are looking for something that is moving toward the ground and slowing down. This could be a diver, right after she hits the water, or a parachutist right after the chute is opened. Try to think of another example.

40. Working It Out: Acceleration
A car traveling along a straight highway at 40 mph speeds up to 60 mph during a time interval of 20 s. What is the car’s average acceleration?
Using the symbols vi and vf to represent the initial and final velocities, we have
The car accelerates at 1 mph/s; that is, during each second, its speed increases by 1 mph.
If, on the other hand, the car made this change in velocity in 10 s, our new calculation would yield an average acceleration of 2 mph/s.
These calculations illustrate that acceleration is more than just a change in velocity; it is a measure of the rate at which the velocity changes.

41. A First Look at Falling Objects
With these few ideas, we can now look at a common motion: a ball falling near Earth’s surface. How does the ball fall?
Does it fall faster and faster until it hits the ground?
Or does it reach a certain speed and then remain at that speed for the duration of its fall?
Does the rate at which it falls depend on its weight? For example, would a cannonball fall faster than a feather?
Questions about this type of motion have fascinated scientists since at least the time of Aristotle (4th century BC).
They turned out to be quite difficult to answer.
In fact, modern answers to these questions were not given until early in the 17th century.

42. A First Look at Falling Objects
Until that time, the accepted answers were those attributed to Aristotle, who believed the following:
Every motion required a mover (object) and a goal toward which the object moved.
An object falling from a height to its natural resting place on an immobile Earth—its goal—would would travel with a speed determined by its weight divided by the resistance of the medium through which it traveled.
Heavier objects would naturally travel faster than light ones.
Much heavier objects would presumably fall even more rapidly.
A cannonball would fall more slowly in molasses than in air.
A feather would flutter down in air, but would float in molasses.
In Aristotle’s view, all change was motion. Falling bodies were just one example of change—a change of place.
Changes of temperature, color, or texture were other examples of motion in Aristotle’s world view.

43. A First Look at Falling Objects
The advantage of Aristotle’s system was that he dealt with concrete, observable situations that we encounter every day.
This advantage led to serious and rewarding discussions over the next 1500 years.
Great scientists such as Galileo carefully analyzed Aristotle’s ideas, which included a prediction that a 10-pound rock should fall significantly faster than a 1-pound rock.

44. A First Look at Falling Objects
You can perform an equivalent experiment to test the theory.
Hold a heavy book (a physics text is quite appropriate) and a piece of paper at equal heights above the floor and drop them simultaneously.
Which falls faster?
Now repeat the experiment, but this time wad the paper into a tight ball.
How do the results differ?
In the first case, the book fell much faster than the paper.
This is in qualitative agreement with Aristotle’s claim.
But the second case certainly disagrees with Aristotle.

45. A First Look at Falling Objects
Using the Aristotelian rule to predict the fall of the paper and book conflicts with reality.
If the book is significantly heavier than the paper, according to Aristotle, the book should drop at a speed significantly faster than the paper.
This means that the book would hit the floor well before the paper!
Clearly, Aristotle was wrong.

46. A First Look at Falling Objects
Our experiment with the book and paper might lead you to believe that in the absence of any resistance, as, for example, in a vacuum, two objects would fall side by side, independent of their weights.
This means that a cannonball and a feather would fall together in a vacuum.
This was the opinion of Galileo Galilei, an Italian physicist of the 17th century.

47. A First Look at Falling Objects
Galileo is often called the founder of modern science, as much because of his style of building a physics world view as because of his particular contributions.
His style was characterized by a strong desire to verify his theories with measurements; that is, he performed experiments to check his ideas.
His goal was simple: to find rules of nature—often in the form of equations—that expressed the results of his investigations.
His work led to some new ideas about motion.
He developed the concepts and mathematical language necessary to describe motion.

48. A First Look at Falling Objects
For example, he invented the concept of acceleration.
Although Galileo was unable to directly test his ideas about free fall, he did suggest the following thought experiment.
Imagine dropping three identical objects simultaneously from the same height.
The Aristotelians would agree that the three would fall side by side.

49. Free Fall: Making a Rule of Nature
Using modern techniques we can take a strobe photograph of a falling ball and “see” its motion.
The ball is clearly not moving at a constant speed.
We can tell this by noting that the distances between the images are continually increasing.
As Galileo showed, the ball falls with a constant acceleration.

50. Free Fall: Making a Rule of Nature
Constant acceleration means that the speed changes by the same amount during each second.
If, for example, we find that the ball’s speed changed by a certain amount during 1 second at the beginning of the flight, then it would change by the same amount during any other 1 second of its flight.
We now know that for the case of the vertical ramp (free fall), the acceleration is about 9.8 (meters per second) per second (32 [feet per second] per second).
This value is known as the acceleration due to gravity and varies slightly from place to place on Earth’s surface.

51. Free Fall: Making a Rule of Nature
Students at Montana State University in Bozeman have determined that the value of the acceleration due to gravity in the basement of the old physics building is (meters per second) per second.
For convenience in calculations we will usually round off this value to 10 (meters per second) per second.
At any time during the fall, if we know its speed and its acceleration, we can calculate how fast the ball will be moving 1 second later.

52. Free Fall: Making a Rule of Nature
Assume that the ball is traveling 40 meters per second with an acceleration of 10 (meters per second) per second.
This acceleration means that the speed will change by 10 meters per second during each second.
Because the ball is speeding up, 1 second later the ball will be traveling with a speed of 50 meters per second.
One second after that, the speed will be 60 meters per second, and so on.

53. Free Fall: Making a Rule of Nature
As a final example of free fall, consider a thrill-seeking skydiver who jumps from a plane and decides not to pull the parachute cord until 30 seconds have elapsed.
Our diver accelerates at a rate of 10 meters per second each second.
Thus, at the end of ½ minute (assuming our skydiver can resist pulling the cord for that long), the speed will be about 300 meters per second. That is about 675 mph!
Actually, as we will see in Chapter 3, this description of free fall is quite inaccurate when the effect of air resistance becomes important.

54. Working It Out: Should You Jump?
You are standing at the top of a waterfall looking down at a deep pool of water below.
Your friends think it is safe to jump, but you are worried that you may be too high for comfort.
You pick up a rock and drop it into the pool.
You count “one-one thousand, two-one thousand, three-one thousand,” and find that the rock takes about 3 s to fall.
How fast would you be going just before you hit the water?
Any falling object, a rock or a person, speeds up by 10 m/s for every second that goes by (if we neglect air resistance).
If you drop with an initial speed of zero, you would be traveling 10 m/s after the first second, 20 m/s after the second second, and 30 m/s (nearly 70 mph!) right before you hit the water.
We suggest that you point your toes.

55. Working It Out: Should You Jump?
How tall is the waterfall?
The rock hit the water going 30 m/s, but clearly the rock did not travel this fast for the entire fall.
The rock’s initial speed was zero and increased uniformly to 30 m/s.
The rock’s average speed during the fall was 15 m/s (halfway between zero and 30 m/s), and the rock had this average speed for 3 s.
That means that, on average, the rock fell 15 m every second for 3 s, for a total of 45 m.

56. Starting with an Initial Velocity
What happens if the object is already in motion when we start our observations?
Suppose, for example, a ball is thrown vertically upward.
Our experience tells us that it will slow, stop, and then fall.
Examining strobe photographs of objects thrown vertically upward shows that the behavior of the rising object is symmetrical to that of the same object falling.
The speed changes by 10 meters per second during each second.
In fact, the strobe photograph could just as well have been a photograph of a ball rising.

57. Starting with an Initial Velocity
We can use the symmetry between motion vertically upward and downward in answering the following question:
if you throw a ball vertically upward with an initial speed of 20 meters per second, how long will it take to reach its maximum height?
Ignoring air resistance, we know that the ball slows down by 10 meters per second during each second.
Therefore, at the end of 1 second, it will be going 10 meters per second.
At the end of 2 seconds, it will have an instantaneous speed of zero.
Therefore, the ball takes 2 seconds to reach the top of its path.

58. On the Bus
Q: If you could throw the ball with a vertical speed of 40 meters per second, how long will it take to reach its maximum height?
A: The ball will take 4 seconds to reach its maximum height.

59. A Subtle Point
Let’s pause to emphasize the fact that Galileo used experiment and reasoning to discover a pattern in nature.
He could discern the motion of objects subject only to the pull of Earth’s gravity.
Using simple mathematics and the rule that he formulated, you can calculate the outcome of future experiments.
In a limited but very real way, you can predict the future.
Predictions based on this rule are not the crystal-ball type popularized in science fiction, but they represent a very real accomplishment.
The discovery of patterns and the creation of rules of nature are central in physicists’ attempts to build a world view.

60. Summary
We began building a physics world view with the study of motion because motion is a dominant characteristic of the universe.
We can obtain data about the motion of objects from strobe photographs.
The average speed of an object is the distance d it travels divided by the time t it takes to travel this distance,
The units for speed are distance divided by time, such as meters per second or kilometer per hour.
Instantaneous speed is equal to the average speed taken over a very small time interval.
Speed in a given direction is known as velocity, a vector quantity.

61. Summary
Displacement is a vector quantity giving the straight-line distance and direction from an initial position to a final position.
Average velocity is the change in position (displacement) divided by the time taken,
Acceleration is the change in velocity divided by the time it takes to make the change,
Acceleration is a vector.
The units for acceleration are equal to those of speed divided by time such as (meters per second) per second or (kilometers per hour) per second.
Galileo reasoned that all objects fall at the same rate in the absence of any air resistance.
Furthermore, he discovered that these free-falling objects fall with a constant acceleration of about 10 (meters per second) per second.