3. One-dimensional motion

We want to understand the meanings of time, distance, speed, velocity, and acceleration. We all have some experience with motion but, as for many aspects of daily life, we have a somewhat vague notion of this common event. We have all driven from our home to a friend’s house but we never had any reason to get our rulers and stopwatches to carefully quantify motion beyond our relative words of fast and slow, moving or stopped. As we measure specific aspects of nature, we are always surprised that it is not exactly what we had assumed. We are surprised by the results of measurements. Science means “facts and understandings obtained from repeatable measurements-billions of measurements have been made.” These are the most important points of the course.

CQ: Write on a sheet of paper to turn in at the end of the class time. Take 15 seconds to answer each of the following questions. What sort of things have you seen in motion? Which things can you see moving right now? Describe “distance.” How much time elapses between heartbeats? How much time is spent shaking your head? Describe something you can do in 1 second. In 10. In 100. For how long does a thrown ball stay in the air? How long does it take you to walk up a flight of stairs? How long does it take a car to traverse a highway on-ramp? How long does it take a rock dropped from shoulder-height to hit the ground? How long will a horizontally-shot bullet stay in the air? Do you believe that the dropped rock and fired bullet both take the same amount of time to fall to the ground? Nature surprises us. We will be surprised in every chapter. In this chapter, we will refine our notion of motion.

Imagine for a moment a world in which nothing is moving-no cars, trains, or planes in motion, not even a strolling animal or a bush blowing in the wind. Just 150 years ago-before we developed engines for trains and such-very few objects were in motion besides people and animals. Since it is only living things that can move themselves of their own accord, about the only other objects in motion were the wind, rising smoke, rivers, clouds, and falling rain. Ask tribes-people why clouds move and they might answer “Because that’s what they like to do.” This makes sense to me. When our earliest natural philosophers pondered the nature of motion, they had only the motions of these few objects to explain. Empedocles (490 - 430 BCE) and Aristotle (384 - 322 BCE) explained these motions in terms of things seeking their own level in matter’s hierarchy of water, earth, air, fire, and heavens.

CQ: Do stars, comets, and planets move? Can you think of other things that move? Do they move of their own accord?

They also asked what is it that temporarily keeps a rock moving after you have thrown it. How would you explain it? Our first moving machinery-sailboats and grinding mills, for example-were pushed or pulled by people, animals, wind, or water. They moved only as something else pushed them. Back around the year 1850, the first time people saw a train moving under its own power they typically said that they were astonished because “It moved just like a living thing.” Before the train, only living things could move under their own power. We are constantly in motion today, driving around town and flying around the continents, but just a couple centuries ago, a busy city was filled only with foot traffic and some horse- or people-drawn carts. The city contained only motionless buildings. In this Wisconsin Historical Society photo, foot traffic in an otherwise motionless city is seen as people walk to work at a Chicago factory in 1886. Copy the following URL into your browser to see the caption for this photo.

http://www.wisconsinhistory.org/whi/fullRecord.asp?id=24888&qstring=http%3A%2F%2Fwww%2Ewisconsinhistory%2Eorg%2Fwhi%2Fresults%2Easp%3Fkeyword1%3D24888%26search%5Ffield1%3Dimage%255Fid%26search%5Ftype%3Dadvanced%26sort%5Fby%3Ddate%26boolean%5Ftype1%3Dand%26boolean%5Ftype2%3Dand

By the way, animals are naturally equipped to detect the motion of other objects because such movement often is of biological consequence. We also quickly learn to ignore those motions which are of no biological consequence, such as the swaying of grass in the wind, though we still might still enjoy such a sight in a mountain meadow. But our animal senses are easily fooled. Akiyoshi Kitaoka from the Ritsumeikan University in Kyoto, Japan demonstrates that sometimes motion is just an illusion. Motion is also art in interactive exhibits, movement, dance, and in kinetic sculptures. Motion can be frozen in time.

Motion is a fundamental aspect of nature, but what exactly is the nature of motion? While driving your car past trees and such, has it ever seemed to you that you are motionless and that instead it is the objects outside the car that are moving past you? From your point of view, are you in motion or are the passing tress and such in motion?

CQ: Are or are not those trees in motion?

Motion involves relative movement. While driving your car for example, you and your passengers are not in motion relative to each other but you are in motion relative to the ground, less so relative to other cars you pass, and about twice so relative to oncoming cars.

Can you be in motion without knowing it? You may have noticed that if you close your eyes while riding in a car, you are unaware of the car’s constant speed but do sense its changes in motion (termed “acceleration”). We feel the acceleration as the car seat pushes us forward. We also feel acceleration as an elevator begins moving and later slows to a stop. But without sensing it at all, we are in motion as the Earth spins in its daily cycle and travels in its annual path around the Sun; at the same time, the Solar System orbits the center of the galaxy, and the galaxies are moving away from each other within the expanding universe. How would you quantify motion?

3.1. The motion experiments of Galileo

Galileo was among the first persons to experimentally study motion. His work began to reveal to us the true nature of motion. He is the hero of this chapter. Galileo (1564-1642) and Shakespeare were both born in the same year. (This is also the year that Michelangelo died.) Some 78 years later, Galileo died during the same year that Newton was born (not fig Newton but Isaac Newton, who is the hero of the next chapter).

Around the year 1590, Galileo 2 began to quantify motion by measuring the time needed for rolling balls to travel given distances along tilted pathways or inclined planes. The steeper the incline, the faster the balls will travel. At the time, there were variously named units of distance-cubits and leagues, for example-and the day had been divided into twenty-four hours but no clocks yet existed that counted the tiny intervals we call seconds. The day was divided only into morning, noon, afternoon, and evening and such. Galileo made his measurements of motion before second-hands existed on clocks. He measured time intervals by counting heartbeats, water drops, and pendulum swings. By the way, what causes the oscillatory motion of a pendulum?

Dropped objects take only a fraction of a second to fall to the ground; it is hard to time such a short interval by counting fractions of heartbeats. Galileo used inclined planes to stretch the time duration of the motion. The less steep the incline, the longer will be the duration of the motion.

Speed represents a distance traveled divided by the time taken to travel that distance. For a modern example, driving at 60 mph means that we traverse a distance of 60 miles in one hour. Galileo quantified motion in terms of distance traversed in each of a series of, for example, three-heartbeat time units. That is, what distance was covered in the first three-heartbeat time period and what distance was covered in the second three-heartbeat time period, and the third, and so on. Galileo defined acceleration to be the change in speed per unit time. (While driving our car, we say that we press the accelerator to speed up.)

CQ: Suppose you begin from a stopped position to drive down a highway on-ramp, you traverse a small distance during the first three-heartbeat time unit (write down an estimate of this distance), a larger distance through the next three-heartbeat time unit (write down an estimate of this distance), and a yet larger distance through the next (write down an estimate of this distance) as your speed increases from slow to fast and then faster.

Galileo found that as a ball rolls down an inclined plane, its speed increases linearly in time – it is accelerating – but its position increases as the square of time. Mathematically, these are written \(v \propto t\) and \(x \propto t^2\). Using the letter ‘a’ as the constant of proportionality, these two relations are also written \(v = at\) and \(x = \frac{1}{2} at^2\).

CQ: Would you have guessed before making these measurements that nature behaves in this specific, mathematical way? Would you expect nature to behave in a mathematical way in any situation?

Luckily, these speed and acceleration equations hold no matter what object-from wooden spheres to metal cylinders-rolls down the inclined plane. They also hold whether the incline is set nearly horizontal or completely vertical, which then results in “free-fall” motion. In the year 1632, Galileo published his results in Dialogue Concerning Two New Sciences. Before Galileo, nobody had stumbled across the importance of the time rate of change of speed. Galileo quantified the change in speed, which is acceleration, \(a = \frac{v_f - v_i}{t_f - t_i} = \frac{\Delta v}{\Delta t} = \frac{dv}{dt}\). We will use the Greek letter delta for the symbol to represent “change.” It is always the “final minus initial” values. Just a few decades later, Isaac Newton would build upon Galileo’s work to figure out that forces cause accelerations.

What other sorts of natural phenomena might be describable by an equation? The answer is: all of them. We find out more accurately how nature works when we make measurements and discover the equations that describe those measurements. This is the goal of the physicist. Every scientist is seeking the “mother load,” which is to be the first person to understand a newly-found aspect of nature. Each new understanding clears up previous confusion and opens new doors of inquiry. Remember also that nature nearly always has more imagination then us, we rarely predict ahead of time how nature behaves: we instead go measure nature and are surprised by the result. Science means facts and understandings obtained from repeatable measurements-billions of measurements have been made. These are the most important points of the course.

Lets look one-by-one at distance, time, speed, and acceleration. A goal of this chapter is for us to gain an understanding of the physical meaning of speed and acceleration.

3.2. Time

Time is measured in seconds. We sometimes count seconds by counting out loud: “one thousand, two thousand.” The time that elapses between heartbeats is about three-fourths seconds.

Lifetimes typically last for about one billion heart beats, no matter the species or heart rate. In general, the larger the animal, the slower the heart rate. Small animals having heart rates ten times the human rate and have lifetimes that are one-tenth as long as a human’s lifetime.

Example: How many seconds are in a year?

\[\frac{60sec}{min} \cdot \frac{60min}{hour} \cdot \frac{24hours}{day} \cdot \frac{365days}{year}\]

When estimating calculations, a year has about \(\pi \times 10^7\) seconds.

Example: How many years are there in a billion seconds?

\(\frac{10^9 s}{3.15 \times 10^7 \frac{sec}{year}} = 31.7\) years.

The age of the Earth is 4.5 billion years or \(1.5 \times 10^{17}\) seconds, and the age of the universe is 13 billion years or \(4 \times 10^{17}\) seconds.

CQ: What is the duration of the trajectory of this tossed item? For how many seconds have you lived?

3.3. Distance

Distance is a measure of how far you have traveled and is written with the symbol ‘x’. Distance is measured in meters and sometimes stated in terms of kilometers or miles and such. One mile is 1.6 kilometers. Let’s look at the size of millimeters and centimeters on this meter stick. Let’s look though Jeff Phillips’ list of order-of-magnitude estimates of times, lengths, masses, speeds, accelerations, forces, and energies for various objects. Let’s look at the National High Magnetic Field Laboratory’s visual tour through the powers of ten, from the Universe, to the Milky Way Galaxy, and down to the subatomic universe of electrons and protons. By the way, the NASA Worldwind website lets you zoom from satellite altitude down to any location on the Earth-down to street level for many U.S. cities.

CQ: What is the length of your desk top? What is the width of this room? How far do you live from the classroom?

3.4. Velocity

Velocity is a measure of how fast you are traveling past something else and is written with the symbol \(v\). Speed is measured in \(m/s\) (pronounced meters per second, just as 6 slices of pie per 2 persons is written 6 pieces/2 persons = 3 pieces/person). The speed of a car is 60 mph = 100 km/h. When a car moves at a constant speed-say 20 m/s-its position is changing by a constant amount-20 meters, in this example-every second, even between the seventh and eighth seconds. By the way, fingernails grow and continents drift at a speed of about 2 cm/year. Speed is always measured relative to something else, which in turn may or may not be in motion relative to something else. For example, a child might walk across your lap while you are sitting in a car that is passing another car along a row of houses. Suppose that you stand on a moving continent and point one finger in the direction of the continent’s motion and another opposite its motion. With what speed do the fingernails on these two hands then move?

The definition of velocity is

\[v = \frac{\Delta x}{\Delta t} = \frac{X_{final} - X_{initial}}{t_{final} - t_{initial}} = \frac{x_f - x_i}{t_f - t_i}.\]

Velocity = change in position / change in time. It is a measure of how quickly the position of an object is changing.

The change \(\Delta\) is always “final minus initial.” We often choose subscripts 0 and 1 to represent the two values or drop the “final” subscript on one variable. The numerical value of speed calculated from this equation is independent of the axis origin. One stationary observer might put \(x = 0\) in Los Angeles (Pacific time) while another puts \(x = 0\) in Natchitoches (Central time) but both calculate the same speed. For example, the Natchitoches observer sees a car at \(x_i = 50\) miles at time \(t_i = 5\) hours and then is at position \(x_f = 150\) miles at time \(t_f = 7\) hours. Its speed is then

\[v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i} = \frac{150 - 50 miles}{7 - 5 h} = 50 mph.\]

The same numerical speed is calculated by the observer sitting in Los Angeles, 2000 miles away using clocks shifted by two hours.

\[v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i} = \frac{2150 - 2050 miles}{5 - 3 h} = 50 mph.\]

Physicists require that models of nature be independent of the observers coordinate system and independent of the observer’s system of units (this is gauge theory).

CQ: Use the ruler and stop watch to measure your own walking speed or the speed of a bug or fly. (By the way, how does a fly go about landing on the ceiling? Does it first invert while flying and then land while inverted?)

Example: To convert speeds from mph to m/s, multiply by 0.44, or to convert m/s to mph multiply by 2.25. For example, 10 m/s = 22.5 mph.

\[\frac{1 mile}{hour} \cdot \frac{1.6 km}{mile} \cdot \frac{1000m}{km} \cdot \frac{1 hour}{3600s} = 0.44 \frac{km}{s}\]

CQ: Show that 60 mph = 88 ft/s. This means that 600 mph = 880 ft/s.

\[\frac{60 miles}{hour} \cdot \frac{5280 ft}{mile} \cdot \frac{1 hour}{3600s} = 88 \frac{ft}{s}\]

Example:

What is the speed of blood flowing in your arm? When I plug and release the vein in my arm here, I see blood move 0.05 m in 0.25 s. Its speed is then \(v = \frac{\Delta x}{\Delta t} = \frac{0.05 m}{0.25 s} = 0.2 \frac{m}{s}\).

CQ: If a dog runs 100 meters in 20 seconds then its speed \(v = \frac{\Delta x}{\Delta t} = 100 m / 20 s = 5 m/s\)

Example:

The speed of light is \(3 \times 10^8 \frac{m}{s}\). At this speed, how many times, \(n\), in one second would light travel a distance equal to the circumference of the Earth? The radius of the Earth is \(r_e = 6.37 \times 10^6 m\) and its circumference is H = 2pi r = ( 2 )( 3.14 )( 6.37 times 10^6 m ) = 4.00 x 10^7 m. The distance light moves is given by \(x = vt = ct = nH\). Solve for \(n = \frac{ct}{H} = \frac{3 \times 10^8 m/s \cdot 1 s}{4.00 \times 10^7 m} = 7.5\).

Distinguish average from instantaneous speed: get into your car, turn it on, speed up, drive 10 miles at 60 mph, passing at 70 mph and slowing to 40 mph for animals, then stop and get out. Picture this motion by putting yourself into that car making that trip. You will have an “average speed = total distance traveled /total time” but at any instant your “instantaneous speed” might have been any number between 0 and 70 mph.

We have a speed if our location is changing, otherwise we are not moving. What is the difference between speed and acceleration? We have an acceleration if our velocity is changing.

3.5. Acceleration

Acceleration is a measure of how quickly a speed is changing. It is written with the symbol ‘a’ and is measured in \(\frac{m}{s^2}\) (pronounced meters per second squared). Moving at constant acceleration means your speed increases by a constant amount per unit of time. The definition of acceleration is

\[a = \frac{\Delta v}{\Delta t}\]

Acceleration = change in speed / change in time. It is a measure of how quickly the speed of an object is changing.

Here and here are some sample values of acceleration.

Example: A person jumps off a platform and experiences a vertical increase in speed of 4.9 m/s in 0.5s. The acceleration of the person is

\[a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i} = \frac{4.9 \frac{m}{s} - 0 \frac{m}{s}}{0.5s} = 9.8 \frac{m}{s^2}\]

We use the symbol ‘g’ to represent this acceleration due to gravity.

Example: It takes the Shuttle 50 seconds to reach a speed of 750 mph, 1 minute 47 seconds to reach a speed of 2600 mph, and 8.5 minutes to reach 17,000 mph. What is its average acceleration at each of these three points? We multiply 750, 2600, and 17,000 mph by 0.44 to obtain 333.3, 1156, and 7560 m/s. The acceleration is then \(a = \frac{\Delta v}{\Delta t} = \frac{333.3 m/s}{50 s} = 6.7 \frac{m}{s^2}\), which is 6.7/9.8 = 0.7g, \(a = \frac{\Delta v}{\Delta t} = \frac{1156 m/s}{60 + 47 s} = 10.8 \frac{m}{s^2}\), which is 10.8/9.8 = 1.1 g, and \(a = \frac{\Delta v}{\Delta t} = \frac{7560 m/s}{8.5 \times 60 s} = 14.8 m/s^2\), which is 14.8/9.8 = 1.5g. For a period just before the engines are cutoff, its acceleration grows to 3g.

We have no sensation of speed but we feel acceleration in an elevator as it starts or stops. As you push the “accelerator pedal” to speed up to get onto the highway, you see the speedometer reading increase and you feel the acceleration as your back is pressed into the car seat, making you feel “heavier.” (On page 44 of The Sciences, An Integrated Approach (James Trefil and Robert M. Hazen, 2004, John Wiley & Sons, Inc., Trefil describes his experience of g-forces as being similar to having multiple persons sitting on you all at once. Trefil and jet pilots say that a 5g acceleration feels like five people sitting on top of you.) If this press stays constant then the acceleration is constant. If you press and release the peddle several times, you are thrown into and out of the seat and will feel the acceleration change in an irritating way. Can acceleration change? Yes, but not in this course. We want to understand the meaning of acceleration and not bury it for no reason with complicated mathematics. If acceleration changes then the motion equations given below do not apply; instead, calculus must be used to obtain other forms of these equations. The time rate of change of acceleration has been aptly named “jerk.” This is of interest in the design of public transportation vehicles but very few natural phenomenon involve the rate of change of acceleration. While driving our cars, we might accelerate gently and feel a gentle push, or press the pedal to the floor to produce a high ‘a’ and feel strongly pressed into the chair of the car. By the way, the time rate of change of a is called jerk–because we do feel a jerking motion. You’ll feel \(\frac{da}{dt}\) if you quickly press and release the gas pedal several times.

3.6. Acceleration is caused by force

In everyday life, we describe a force as a push or a pull. Isaac Newton, who will be the hero of the next chapter, combined and extended the motion studies of Kepler, Galileo, and others by stating that force equals mass times acceleration, \(F = ma\). Mass \(m\) resists the acceleration. The gravitational force between you and the Earth results in your weight, \(W = mg\). When you drop an object, its weight is the force that causes its acceleration. We have \(F = ma = mg\). We cancel the \(m\) and see that all weights fall with the same acceleration, \(a = g\), as was Galileo’s rumor. When the force is constant then the acceleration is constant, and we have \(a = F/m\).

3.7. The motion equations for constant acceleration

For constant acceleration, the average speed occurs at the midpoint of a time interval

\[v_{ave} = \frac{v + v_0}{2}.\]

(This is seen in the graph of a straight line. We use this fact in labs involving spark tape).

Let’s set \(t_f = 0\) and drop the subscript on \(t_i\) so that \(v_{ave} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}\) becomes

\[v_{ave} = \frac{x_f - x_i}{t},\]

which can be written as \(x_f = x_i + v_{ave} t\), or, since one subscript is easier than two,

\[x = x_0 + v_{ave} t.\]

Notice that we can solve this for \(t\) to get \(t = \frac{v - v_0}{a}\).

Substitute this \(v\) in the equation \(x = x_0 + v_{ave} t = x_0 + (v + v_0)t/2\) to get

\[x = x_0 + \frac{(v_0 + at + v_0)t}{2} = x_0 + \frac{(2v_0 + at)t}{2} = x_0 + v_0t + \frac{1}{2}at^2.\]

If instead of substituting \(v\) in the above equation, we can instead substitute \(t\) as noted to get,

\[x = x_0 + \frac{(v + v_0)t}{2} = x_0 + \frac{(v + v_0)(v - v_0)}{2a}\]

or

\[v^2 = v_0^2 + 2a(x-x_0).\]

Galileo’s measurements of motion down inclined planes resulted in the following so-called motion equations that we use to describe motion under constant acceleration.

1. \(x = x_0 + v_0 t + \frac{1}{2} at^2\) (notice that this is a quadratic equation in t and so has two solutions)
2. \(v = v_0 + at\)
3. \(v^2 = v_0^2 + 2a(x-x_0)\)
4. \(x = x_0 + \frac{1}{2}(v + v_0)t\)
5. \(\bar{v} = (v_0 + v) / 2\) (for linear relations, the average occurs at the midpoint of an interval).

Notice that when \(x_0 = 0\) and \(v_0 = 0\) these equations become \(x = \frac{1}{2}at^2\), \(v = at\), and \(v^2 = 2ax\).

This looks like five equations but it is really only two independent equations plus three rearrangements. For these two equations, we can have only two unknowns.

It does not matter what is the object-from falling coins to rocket ships-the motion of any object moving with constant acceleration can be described by these motion equations. In these equations, \(x\) represents the distance an object has moved in time \(t\) from its starting place, \(x_0\). The object’s speed is represented by \(v\) and its initial speed by \(v_0\). Speed is visualized in terms of how quickly an object’s position is changing. Notice that \(x\) represents how far an object has moved while \(v\) represents how fast it is moving or how quickly the object’s position is changing while acceleration represents how quickly its speed is changing in time. Speed is the time rate of change of position while acceleration is the time rate of change of speed. In symbols, speed \(v= \Delta x / \Delta t\), where the change in position is \(\Delta x = X_{final} - X_{initial}\), and \(a = \Delta v / \Delta t\). (In calculus, differential changes are used, making \(v = dx/dt\) and \(a = dv/dt = d^2x/dt^2\).)

Example:

Suppose a car is accelerated from a stopped position to a speed of 20 m/s (45 mph) in 10 seconds. Its acceleration is \(a = \Delta v / \Delta t = ( 20 m/s ) / 10 s = 2 m/s^2\) and it will have traveled a distance

\[x = \frac{1}{2}at^2 = ( \frac{1}{2} )( 2 m/s^2 )( 10s )2 = 100 m (109 yards).\]

Compared to your own driving style, does this seem like a high or low rate of acceleration? Do you typically accelerate from 0 to 20 m/s (45 mph) in 10 seconds while traveling just 100 meters?

3.8. Calculus topic: obtaining the constant acceleration equations

For differential changes in speed, \(a = dv/dt\) or \(v = \int a \, dt = a \int \, dt = at + c\). The integration constant is determined from the condition that \(v(t=0) = v_0\), giving \(v = v_0 + at\). From \(v = dx/dt\) or \(x = \int v \, dt = \int (v_0 + at ) \, dt\), we integrate to get \(x = v_0 t + \frac{1}{2} at^2 + c\). The constant is found from \(x(t=0) = x_0\). Writing \(a = dv/dt = dx/dt dv/dx = v dv/dx\) and integrating from \(x_0\) to \(x\) and \(0\) to \(t\) we get \(\int v \, dv = \int a \, dx\) or \(v^2 = v_0^2+2a(x-x_0)\).

3.9. Area under the curve

Suppose an object travels through a first time interval at constant speed, then travels through a second and equal distance at a second speed, and then a third and so on. The total distance traveled is

\[\Delta x = v_0 \Delta t + v_1 \Delta t + v_2 \Delta t + v_3 \Delta t + v_4 \Delta t + v_5 \Delta t + v_6 \Delta t + \ldots + v_n \Delta t = \sum v_i \Delta t = \int v \, dt.\]

An Example with a Caption