Chapter 2: Average Speed and Velocity

https://www.mnn.com/green-tech/transportation/stories/how-fast-could-you-travel-across-the-us-in-the-1800s

2.1 Average Speed & Velocity ^3, 4

Speed and velocity are both measures of how quickly something (car, person, jet, etc.) moves from one place to another.  While we often run into people using these interchangeably, they are different measures of the same phenomena and require you to look at the same problem in different ways.

Suppose you ask friends to help move a heavy refrigerator roughly 5 m, and the reward will be pizza and beer afterwards.  These friends might think this was an excuse to get several of them together but then one important detail is left out. This refrigerator must be moved to an apartment directly above the one it is in.  The friends soon understand that the total distance the refrigerator is to be moved is a lot greater than the roughly 5 m.  In this instance, the refrigerator is to be moved about 40 m down the hallway, to the stairs where it is lifted up 5 m vertical using stairs to the second floor and then moved 40 m down the hall into the other apartment.

The original friend was not being dishonest about moving the refrigerator 5 m, as it was technically moved 5 m straight upwards. This 5 m straight upwards would be measured as a displacement.  The total distance travelled in this example would be 40 m + 40 m + 5 m +, the distance up the flight of stairs.

Measures of distance moved are thus defined as the total distance travelled where the change in displacement is the change in position that includes the direction between these two positions.

These measures are also classified as scalar and vector measures with the scalar just being a number of how much and the vector (it generally has an arrow over the symbol) giving a change in the measure and including the direction involved.

Because of this, you will encounter two sets of measures in introductory physics, a scalar and a vector that can be used to measure the same phenomena.  We start this off by looking at the average speed and the average velocity of an event.

[latex]\overrightarrow{v}[/latex] is the Velocity, commonly measured in Metres/Second (m/s) or Kilometres/Hour (km/h) and includes a direction

[latex]\Delta d[/latex] is the Distance Travelled, commonly measured in Metres (m), Kilometres (km)

[latex]\Delta \overrightarrow{d}[/latex] is the Change in Displacement, commonly measured in Metres (m), Kilometres (km) and includes a direction

[latex]t[/latex] is the Time, commonly measured in second (s) or hours (h)

Often [latex]\textit{speed and velocity}[/latex] involve combining several different motions where one is required to calculate the average speed or velocity of some grouping of linked motions.

[latex]Scalar[/latex]  is the amount or how much of a measure and is only a magnitude without other data.

[latex]Vector[/latex]  is a measure that also has a direction associated with the measure. It typically refers to magnitude and directional change from the initial to the final measurement. ⁶

Two of these measures, Time and Distance/Displacement, are classified as base units in the International System of Units⁷.

[latex]\textit{Time prior to 1900}[/latex], was defined as the fraction 1/86400 of an earth day (24 h x 60 min x 60 s). This was shortly redefined (1956-1967) as the fraction 1/31 556 925.9747 of the tropical year of 1900, January 0 at 12 hours ephemeris time. Time is currently defined (1967) by the energy emitted/absorbed by the caesium-133 atom.

Depending on what fields you study in the future, you will encounter numerous measures of time⁸, the largest of which is the galactic year, which is the time based on one complete rotation of our galaxy. One galactic year is currently given as 230 million years.

Historically, the measurement of time presented early scientists with a number of challenges. Originally the Egyptian calendar of one year was defined by measurements based on the heliacal rising⁹ of the binary star Sirius, marking the onset of the flooding of the Nile river. Many ancient civilizations, including the Babylonians, Sumerians, Greeks and Maori, based their calendars on the phenomenon of heliacal rising. The value of heliacal rising allowed the correction of early calendars’ lack of a leap year to be corrected. This was replaced with the current Julian (proposed by Julius Caesar in 46 BC) and Alexandrian (based on the early Egyptian calendar) calendars. Currently the Gregorian calendar, a reformation of the Julian calendar, is the one most commonly used.

The hourglass and the water clock¹⁰ were early measuring devices of time. The hourglass (sandglass, sand timer, sand clock or egg timer) is recorded in history as far back as 1600s BC in Babylonia and Egypt. These early hourglasses were two joined glass bulbs, one filled with sand that flowed freely from one bulb to the other when the sand-filled bulb was positioned above the other. While common throughout the ancient Middle East, records of its arrival in Europe indicate its presence there from the mid-1300s. The hourglass remained in common use but started to decline in popularity as a timing device in the 1500s, due to the technological development of mechanical clocks.

Water clocks are claimed to have been present in China as early as 4000 BC, and documented in Egypt around the 1600s BC. The design of a water clock is simple; water flows slowly from an upper source to a container where the amount of water, and hence time, can be measured. Hourglasses and water clocks depended on the rate of flow of either sand or water, which indicated the passage of time.

Galileo Galilei created a time-measuring device called the pulsilogon, which he based on the regularity in the time of a swinging pendulum. One of his first applications of the pulsilogon was to measure a person’s pulse. This was accomplished by lengthening or shorting the pendulum until the time for one swing of the pendulum (called the period) matched the rate of the pulse of the person. Medical historians view this invention as the birth of precision in medicine. Using this pendulum time piece, Galileo was able to discover some of his foundational concepts in physics¹¹.

Length prior to 1793 was defined by the fraction 1/10 000 000 of the meridian length (through Paris) between the North Pole and the Equator. This also was redefined for a short time (1960-1983) by the wavelength of radiation emitted by the Krypton-86 atom. Length is currently defined (1983) by a Speed Equation: the distance that light travels in a vacuum in a fraction of 1/299 792 458 seconds.

Using the speed equation: [latex]v =\dfrac{d}{t}[/latex], we can see that [latex]d =\textit{ v t}[/latex] which means for the base measure of length:

One meter (m)  =  (speed of light) (1/299 792 458 seconds).

Historically, the measurement of length originates from multiple cultures and civilizations. Common English measures such as the inch (length of a thumb), the foot (the length of a foot), the yard (length of an average pace) and the mile (one thousand of the Roman two steps to a pace) all have historical origins. Most common of all these measures for length was the cubit, which was measured as the distance between the elbow to the tip of the middle finger.  As you can remember from Chapter One, there are many measures used for length, each having specific uses for various trades and specialties.

To get around the confusion of different measures of length and other measures, the metric system of measures was created during the French Revolution in 1799.  The early metric system, based on a decimal fraction of the distance from the Equator to the North Pole, was also used to define the kilogram, i.e., the mass of water contained in a cube of 10 cm in length (decimetre). As time passed, the need for greater accuracy defined the metre and kilogram from two standard artifacts created out of platinum and iridium. These constructions were used to define length and mass, having since becoming common measures in all countries except the United States, Myanmar (Burma) and Liberia. The image to the right is the only surviving artifact of the original metre, and is located at 36, rue Vaugirard, France.

The following example will help to explain the difference in measuring speed and velocity.

∆d = 6 km + 5 km + 5 km + 5 km  

∆d = 21 km 

t = 3 h + 2.5 h + 3 h + 2 h  

t = 10.5 h 

Final change in displacement = 0 km. The hikers ended up in the same position that they started from.

In real life, you often encounter motion on, in or on top of some moving medium.  Common examples of this are one walking on an escalator, a boat moving on a stream or a river, or an aircraft that is flying into moving air, such as the jet stream. 

The simplest way to understand this is to relate to a common experience that most of us have had: walking in or opposite to the direction that an escalator is moving.  

When walking in the same direction as the escalator is moving, you travel faster and only need to add the speed of the escalator to how fast you are walking to calculate your relative speed. An equation for this looks like: 

^vrelative = ^vescalator + ^vwalking

When walking in the opposite direction to the one in which the escalator is moving, you travel more slowly and can simply subtract the speed of the escalator from the speed you are walking to calculate your resulting speed. An equation for this looks like: 

^vrelative = ^vescalator + ^vwalking

Should you not walk fast enough, you could easily end up moving backwards, even though you are walking forwards. 

This is slightly more complicated for aircraft and watercraft that are moving in air or water that itself is moving. If moving in the same direction or the opposite direction that the air or water is moving, one generally adds or subtracts these two velocities as we did above for walking on an escalator. If moving at an angle to the direction the air or water is moving, then generally you draw a sketch and use trigonometry to find the relative velocity. 

This is done using either right angle trigonometry and the Pythagorean Theorem or by using non-right angle trigonometry. The equations to use for these are: 

For any right triangle a, b and c: 

a² + b² = c²

For any non-right triangle a, b and c: 

a² = b² + c² – ²bc cos A 

                                         b² = a² + c² – ²ac cos B

c² = a² + b² – ²ab cos C

Where:

Basic Trigonometric Ratios

[latex]Sin =\dfrac{Opposite}{Hypotenuse}[/latex]     [latex]Cos =\dfrac{Adjacent}{Hypotenuse}[/latex]     [latex]Tan =\dfrac{Opposite}{Adjacent}[/latex]

         Trigonometric Defined Sides

The Ptolemaic system of understanding the motions of the universe survived for the next fourteen centuries until Nicolaus Copernicus (1473-1543) challenged this model in 1530 by placing the Sun in the center of the solar system, orbited by the planets (heliocentric). The arguments against a heliocentric model mainly stemmed from a blend of Christian beliefs and a failure to understand relative motion. People believed that the stresses of a rotating Earth would cause it to tear apart, flying in different directions, and that any object tossed up into the air would land quite far away, opposite to the direction of the Earth’s spin. 

The reigning thought was that man was God’s greatest creation and therefore must be at the center of the universe. Copernicus raised concerns over the Aristotelian concept of why only the Earth would be comprised of the elements of earth, water, air and fire, yet all objects in the heavens were composed of the quintessence.

While Galileo was a gifted and curious scientist, he found himself with financial burdens¹⁵ that at the time outstripped his salary from the University of Padua. His understanding and refinement of the telescope, turning it from 3x magnification to over 9x magnification, won him immediate praise and financial reward from military leaders at the time.  This advancement gave the military defenders of Venice¹⁶ an extra two or more hours’ warning of advancing ships before people on the ships could see the city. Galileo was granted tenure at the university for this advancement, with an increased salary. 

Some of Galileo’s first discoveries with his improved telescope included mountains on the Moon and, on January 7, 1610, four moons on the planet Jupiter¹⁷.  Understanding the risks of his discovery, Galileo name the moons of Jupiter after members of the Medici family, Dukes of Tuscany, gifting the Duke with a further improved 20x magnification telescope to better be able to see the moons named after his family members.

Despite this action, Galileo was warned not to venture outside the protection of the city-state of Venice, which wanted to tax the Church, whose interpretation of scripture defined Man as the centre of a universe that orbited the Earth (geocentric). In 1616, the Catholic Church condemned the Copernican system (Protestant churches of the time allowed for the freedom to interpret the Bible) and Galileo’s stance on the heliocentric universe now placed Galileo in danger of heresy. Galileo increased this risk by not only continuing to defend his position but writing of his beliefs in a work called a “Dialogue Concerning the Two Chief Systems of the World-Ptolemaic and Copernican.”  So confident was he in the validity of his work, Galileo created characters in his book, including one named Simplicio, who clearly represented the current Pope, who discussed the merits of his position.

Galileo’s book was published in March 1632. Five months later, the order to cease publishing was accompanied by an order for Galileo to stand trial. In 161618, when Galileo was threatened with torture in the dungeons below the Vatican, he was given a tour of these torture chambers. He agreed to tone down the writings in his Dialogues, and spent his remaining years in penance under house arrest at his villa until his death in 164219. Perhaps what is unusual about Galileo’s sentence is that the Vatican ordered it to be widely published in scientific circles. A number of historians have argued that the strong stance taken by the Catholic church at the time effectively discouraged the development of science in Italy for many generations afterwards. 

Galileo’s Two New Sciences

The Discourses and Mathematical Demonstrations Relating to Two New Sciences, Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze (1638), was Galileo’s final book, written while he was under house arrest. Following the publication of his Dialogue Concerning the Two Chief Systems of the World-Ptolemaic and Copernican, the Vatican banned any further publication of any of Galileo’s works, including what he would potentially write in the future. This ban resulted in failed attempts to publish his last work in France, Poland and Germany, but it was published by Lodewijk Elzevir in Holland. Galileo seems to have escaped paying a price for this new crime, since all 50 of the Two New Sciences books imported into Rome quickly sold. 

While Two New Sciences was written in a similar form to his earlier work, Two Chief Systems, the discussions between the familiar Simplicio, Sagredo and Salviati, Simplicio no longer held the simplistic, stubborn Aristotelian beliefs of the time, and instead represented Galileo’s early beliefs as he started the studies. Sagredo represented Galileo’s changing views and Salviati represented Galileo’s final views on his research. Such a change in his writing could also have influenced Galileo’s escaping any further punishment outside of house arrest. The importance of this work and Galileo’s trials expose the conditions and the world in which these early researchers worked, and Galileo’s work to understand relative motion.   

Galileo explained why an object on Earth falls straight even though the Earth spins, through his theory regarding objects on a moving ship. Aristotelian arguments against the rotation or spin of the Earth suggested that an object dropped at the equator would land far behind the position from which it was dropped. For instance, if the Earth is rotating at 1000 km/h or 278 m/s at the equator and a dropped object takes two seconds to reach the ground, the object would land 556 m behind; an object taking 4 s to reach the ground would fall behind over a kilometre.  

Galileo compared what was expected to happen according to Aristotelian logic to what can actually be observed in an object dropped on a moving ship. Even without conducting the actual experiment, anyone working on ships knew that objects fall straight down²⁰.  Galileo also proposed that an observer from shore would see a projectile type motion.  In this example, Galileo explained that objects in motion on moving surfaces would be observed differently by an observer on the moving surface and one standing on a surface that is stationary. 

The following examples explore the algebra of relative motion.

1. Mechanical Universe – Episode 5 – Vectors: https://www.youtube.com/watch?v=MWrOozSSdW8

2. Image is from “How fast could you travel across the U.S. in the 1800s?” Source: https://www.mnn.com/green-tech/transportation/stories/how-fast-could-you-travel-across-the-us-in-the-1800s

3. Extra Help – Distance and Displacement: https://www.physicsclassroom.com/class/1DKin/Lesson-1/Distance-and-DisplacementExtra Help – Speed and Velocity: https://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity

4. Galileo is credited as being the first to measure speed as the ratio of distance travelled to the time taken.

5. Average speeds and velocities are generally different than the speed and velocity at some instant.

6. Extra Help – Scalars and Vectors: https://www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors

7. The other fundamental measures are:• Kilogram (kg) is the base measure used in quantifying mass• Ampere (A) is the base measure used in quantifying electric current• Kelvin (K) is the base measure for quantifying temperature• Mole (mol) is the base measure used in quantifying the number of atoms in a substance • Candela (cd) is the base measure used in quantifying the luminous intensity of any light source

8. https://en.wikipedia.org/wiki/Unit_of_time

9. Heliacal rising refers to the moment that a star becomes briefly visible on the Eastern horizon at dawn, having not been visible during the previous year.

10. The sundial while known at the same time as these two clocks is not covered in this Chapter.

11. Article to Read – Big Bang may have created a Mirror Universe where time runs backwards: https://www.pbs.org/wgbh/nova/article/big-bang-may-created-mirror-universe-time-runs-backwards/?fbclid=IwAR1iKXC6uzGJIHEjIwcItxu2q155_FmjqvmqPi9kJ1rEM0i9MMJ431ueuLs

12. Extra Help – Relative Velocity and Riverboat Problems: https://www.physicsclassroom.com/class/vectors/Lesson-1/Relative-Velocity-and-Riverboat-Problems

13. Galileo is credited with developing the concept of relative velocity in his early experiments that he designed to prove the Earth was a sphere and that it rotated around the Sun. It was his publishing of this in his work “Dialogues Concerning the Two Chief World Systems”. Source: http://www.physicscentral.com/explore/plus/galilean-relativity.cfm

14. Article to Read – Was Giordano Bruno Burned at the Stake for Believing in Exoplanets?: https://blogs.scientificamerican.com/observations/was-giordano-bruno-burned-at-the-stake-for-believing-in-exoplanets/

15. Galileo was the oldest son in his family and found himself responsible to raise the dowries of his younger sisters and of the financial care of three children by his mistress, Marina Gamba.

16. Venice at the time of Galileo was rather a boisterous city with a population of around 150,000 people consuming in excess of 40 million bottles of wine each year.

17. These two discoveries acted to destroy the concept of heavenly bodies made of the quintessence. Galileo’s later discovery that the Sun had spots on it further tarnished the image of the perfection of the quintessence.

18. In 1616 the Catholic church banned the 1543 book by Nicholas Copernicus “On the Revolutions of the Celestial Spheres,” which contained the theory that the Earth revolved around the sun.

19. It is worth noting that it was understood that Galileo’s punishment reflected his behavior to wards authority and not of his scientific achievements. As written by a Jesuit scholar of his time: “If Galileo had only known how to retain the favor of the fathers of this college, he would have stood in renown before the world; he would have been spared all his misfortunes, and could have written about everything, even about the motion of the earth.” What should be noted is the the Jesuits were soon teaching that the Earth orbited the Sun as in a heliocentric model in China and other parts of the world.