2.1 Average Speed and Velocity

Speed and velocity are both measures of how quickly something (car, person, jet, etc.) moves from one place to another^[1]. While people often use these interchangeably, they are different measures of the same phenomenon and require looking at the same problem in different ways.

Suppose someone asks several of his friends to help move a heavy refrigerator roughly 5 m, for a reward of pizza and beer afterwards. This may sound straightforward, but 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 described roughly 5 m. In this instance, the refrigerator is to be moved about 40 m down the hallway, up the stairs to the second floor (a vertical distance of 5 m) and then 40 m down the hall into the other apartment.

The friend was honest in their original description of the move: technically, the refrigerator needed to be moved 5 m straight up. However, 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 top 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 phenomenon. We start this off by looking at the average speed and the average velocity of an event.^[2]

[latex]v=\dfrac{\Delta d}{t}[/latex]                           [latex]\vec{v}=\dfrac{\Delta\vec{d}}{t}[/latex]

Where…

• [latex]v[/latex] is the speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{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\vec{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 speed and velocity involve combining several different motions to calculate the average speed or velocity of a grouping of linked motions.

Scalar is the quantity of a measure and is only a magnitude without other data.

Vector is a measure that has an associated direction. It typically refers to magnitude and directional change from the initial to the final measurement.

Two of these measures: time and distance/displacement, are classified as base units in the International System of Units.^[3]

Time

Prior to 1900, time was defined as the fraction, [latex]\tfrac{1}{86\;400}[/latex] of an Earth day (24 h × 60 min × 60 s). This was redefined shortly thereafter (1956-1967) as the fraction [latex]\tfrac{1}{31\;556\;925.9747}[/latex] of the tropical year of 1900, January 0 at 12 hours ephemeris time. Time is currently (since 1967) defined 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 time, 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 rising of the binary star Sirius, marking the onset of the flooding of the Nile River.^[4] Many subsequent civilizations, including the Babylonians, Sumerians, Greeks and the Maori, based their calendars on the phenomenon of heliacal rising, correcting previous calendars’ lack of a leap year. These corrected calendars were replaced with the current Julian (proposed by Julius Caesar in 46 BCE) and Alexandrian (based on the early Egyptian calendar) calendars. Currently the Gregorian calendar, a restructuring of the Julian calendar, is the one most commonly used.

The hourglass and the water clock were early devices that measured time.^[5] The hourglass (sandglass, sand timer, sand clock or egg timer) is recorded in history as far back as 1600s BCE in Babylonia and Egypt. These early hourglasses consisted of two connected glass bulbs, one filled with sand that flowed freely 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 from the mid-1300s onward. The hourglass continued in common usage until the 1500s, when it began to decline in popularity as a timing device due to the technological development of mechanical clocks.

Water clocks are claimed to have been present in China as early as 4000 BCE, and it is known that water clocks were present in Egypt around the 1600s BCE. The design of a water clock is simple: water flows slowly from an upper source to a container where the amount of water and therefore time can be measured. Both hourglasses and water clocks depend on the rate of flow of either sand or water to indicate the passage of time.

The image to the right is of the oldest surviving water clock from Egypt, 1400 BCE. The image below is of water clock calculations by Nabû-apla-iddina (600-500 BCE).

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

Length

The measurement of length originates in 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 concept of two steps to a pace) all have historical origins. Most common of all these measures for length was that of the cubit, which was measured as the distance between the elbow and the tip of the middle finger. As you can remember from chapter one, there are multiples of measures used for length, each having specific uses for various trades and specialties.

The metric system of measures was created during the French Revolution in 1799, to clarify the definition of length and other measures. 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 as the mass of water contained in a cube of 10 cm length (decimetre). As time passed, the need for greater accuracy arose, resulting in the creation of two standard artifacts, created out of platinum and iridium, for the metre and the kilogram. These constructions were used to define length and mass, having since become 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, Paris, France.

Prior to 1793, length was defined by the fraction 1/10000000 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 the fraction 1/299792458 seconds.

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

One meter (m) = (speed of light) (1/299792458 seconds).

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

2.2 Average Speed/Velocity (Converting Units)

In real life, you will often need to convert from multiple units to a shared common unit to be able to solve a problem. For speed, you may be given speeds in m/s, cm/s, km/h, km/s, length in m, cm, km, light years and time in seconds, minutes, hours, days or years. Make sure that all units are the same measure before you start solving the problem; this will make the task significantly easier.

2.3 Relative Velocity in Two Dimensions

In real life, one often encounters motion on or in a moving medium. Common examples of this include: one walking on an escalator, a boat moving on a stream or river, and an aircraft that is flying into moving air such as the jet stream.

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

When walking in the same direction as the escalator is moving, we travel faster. To calculate our relative speed, we add the speed of the escalator to the speed of our walking pace. If we wish to put this in an equation, it looks like the following:

[latex]v_{\text{relative}} = v_{\text{escalator}} + v_{\text{walking}}[/latex]

When walking in the opposite direction as the escalator is moving, we travel slower. To calculate our relative speed, we subtract the speed of the escalator from the speed of our walking pace. If we wish to put this in an equation, it looks like the following:

[latex]v_{\text{relative}} = v_{\text{escalator}} - v_{\text{walking}}[/latex]

Should we not walk fast enough, we could easily end up moving backwards, even though we are walking forwards.^[9]

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, we generally add or subtract 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 one generally draws a sketch and uses trigonometry to find the relative velocity.

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

• For any right triangle a, b and c:
  
  [latex]a^2 + b^2 = c^2[/latex]

 

• For any non-right triangle a, b and c:
  
  [latex]a^2 = b^2 + c^2 - 2bc \cos(A)[/latex]
  [latex]b^2 = a^2 + c^2 - 2ac \cos(B)[/latex]
  [latex]c^2 = a^2 + b^2 - 2ab \cos(C)[/latex]

Where…

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

The Crimes of Galileo

It is challenging today to find anyone who was not told at an early age that the Earth orbits around the Sun; this is the heliocentric theory. However, if we go back about four centuries, one of the greatest scientists of his time, Galileo Galilei (1564-1642), was threatened with torture and nearly executed for teaching such a doctrine.

Centuries earlier, two Greek philosophers, Eratosthenes (276-194 BCE) and Aristarchus (310-230 BCE), espoused the same theory. Using astronomy and geometry, they obtained reasonably accurate measures of the size of the Earth and the Moon, and calculated the distances from the Earth to the Moon and the Sun. Despite the existence of this theory, Ptolemy (100-170 CE) created an elaborate model of the universe, using what he described as crystalline spheres on which all known heavenly bodies moved. The Ptolemaic system of explaining the motions of the universe survived for the next fourteen centuries.

In 1530, Nicolaus Copernicus (1473-1543) challenged the Ptolemaic model with a heliocentric theory that placed the Sun in the centre of the solar system, orbited by the planets. At the time, people believed that if the Earth rotated, it would fly apart, flinging pieces of the planet far and wide in the opposite direction of the Earth’s rotation. Added to the prevailing Christian doctrine of man as God’s greatest creation, who therefore must be at the centre of the universe, this general lack of understanding of relative motion created opposition to Copernicus’s theory.

Copernicus asked why only the Earth would be comprised of the Aristotelian elements of earth, water, air and fire, when all objects in the heavens were composed of the quintessence.^[10] One of Copernicus’s supporters, the Dominican friar Giordano Bruno (1548-1600), expanded upon the Copernican model, declaring that stars were spread throughout infinite space, and positing the possibility of other inhabited worlds. While Bruno, who was burned at the stake, paid with his life for his belief in the accuracy of the heliocentric model, it was Galileo Galilei’s technological advancement, the telescope, that suggested that the Moon was quite similar to the Earth and therefore the known planets were also like the Earth and not composed of the quintessence.

Galileo’s understanding and refinement of the telescope, turning it from 3x magnification to over 9x magnification, won him immediate praise and financial reward from contemporary military leaders. Using the telescope, the military defenders of Venice^[11] could detect the sails of advancing ships two or more hours before the ships could detect the city. While Galileo was a gifted and curious scientist, his financial burdens^[12] outstripped his teaching salary from the University of Padua. Galileo was granted tenure at the university, with an increased salary, for the advancement of the telescope.

Galileo’s first discoveries with his improved telescope included mountains on the Moon, and on January 7, 1610, the four moons of the planet Jupiter.^[13] Understanding the risks of his discovery, Galileo named the moons of Jupiter after members of the Medici family, the Dukes of Tuscany, and gifted the Grand Duke with a further improved 20x magnification telescope to be able to better see the moons named after his family members. Galileo’s political maneuver protected him in Venice, however, he was warned about venturing outside the city due to tension between the Church and the city-state. Venice was attempting to impose taxes on the Church, whose geocentric interpretation of the scriptures held that Man living on Earth was the centre of the universe.

In 1616 the Catholic Church^[14] condemned the Copernican system, placing Galileo in danger of heresy for his stance on the heliocentric universe. Galileo soon increased this risk by not only continuing to defend his position but writing of his beliefs in a work called, “Dialogue Concerning the Two Chief Systems of the World-Ptolemaic and Copernican.” Confident in the validity of his work, Galileo had the characters in his book, including one named Simplicio, who was clearly named after the current Pope, discuss the merits of his position.

The order to cease publishing Galileo’s book, five months after its publication in March 1632, accompanied the order for Galileo to stand trial. Having been threatened in 1616^[15] with torture in the dungeons below the Vatican, a tour of the torture chambers during his trial helped to convince Galileo to tone down the writings in his Dialogues. He spent his remaining years in penance under house arrest at his villa until his death in 1642.^[16]

Ironically, the Vatican ordered Galileo’s sentence to be published widely in scientific circles. Historians have argued that this strong stance taken by the Catholic Church effectively discouraged the development of science in Italy for many generations afterwards.

Galileo’s “Two New Sciences”

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 any future writing. Galileo wrote his final book, Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638), while he was under house arrest. The Vatican ban resulted in failed attempts to publish his last work in France, Poland and Germany, but Lodewijk Elzevir finally published it in Holland. Only fifty copies were exported to Rome, all of which quickly sold.

While Two New Sciences was written in a similar form to Galileo’s earlier work, Two Chief Systems, the discussions continued between the familiar Simplicio, Sagredo and Salviati. Simplicio no longer had the simplistic, stubborn Aristotelian beliefs of the time, and represented Galileo’s early beliefs as he started his studies. Sagredo represented Galileo’s changing views, and Salviati represented Galileo’s final views on his research. Such a change in his writing may have helped Galileo to escape any further punishment beyond house arrest.

This important work paved the way for those who followed Galileo to understand relative motion.

Relative Motion

Aristotelian theory argues against the rotation or spin of the Earth, stating that an object dropped at the equator would land far behind the position from which it was dropped, e.g.: if the Earth is rotating at 1000 km/h or 278 m/s at the equator and the dropped object takes two seconds to reach the ground, it would land 556 m behind; an object taking 4 s to reach the ground would fall behind over a kilometre.

Galileo used the example of falling objects on a moving ship to explain why a falling object could fall straight down if the Earth was indeed spinning. His thought experiment compared what was expected to happen according to Aristotelian logic, to what can actually be observed to happen. Without actually conducting the experiment, anyone working on ships knew that falling objects fall straight down.^[17] Galileo also proposed that if observed from the shore, the observer 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

Exercise Answers

2.1 Average Speed/Velocity

1. 76 km/h
2. 10.2 m/s
3. 1. 600 km
   2. 32 h
4. 1. 0.42 m/s
   2. 0.12 m/s NW
5. 1.13km²

2.2.1 Average Speed /Velocity (Converting Units)

1. 1 m/s = 3.6 km/h. To change m/s to km/h multiply by 3.6 and to change km/h to m/s divide by 3.6
2. 75 m
3. 101 km
4. 30 km
5. 252 km
6. 305 km

2.2.2 Average Speed and Velocity (Similar to 2.2.1 but these take more time to solve)

1. 3220 km² (≈ 3000 km²)
2. 120 km/h
3. 925 km/h
4. 1. 240 km
   2. 96 km/h
5. Impossible to do
6. 0.57 s
7. 4.0 minutes
8. 1.25 m

2.4 Relative Velocity in One Dimension

1. 104 m
2. 1. 90 km/h
   2. 180 km/h
   3. 142 km/h, [latex]angle = 10^{\circ}[/latex]
   4. 18.1°
   5. 128 km/h
   6. 1.95 hours
3. 1. 6.3 cm
   2. 1800 km
   3. 256 km/h
   4. 1.57 h
   5. 2.81 h

2.5.1 Horse Transportation

1. 7.53 × 10⁸ Horses (753 million horses)
2. 6.03 × 10⁸ hectares

2.5.2 The First Measure of the Speed of Light

1. 2.26 ×10⁸ m/s
2. 2.97 × 10⁸ m/s

2.5.3 The Time Delay in Communication between Earth and Mars

1. 377 s or 6 min 17 s
2. 656 s or 10 min 57 s
3. 2642 s or 44 min 2 s

2.5.4 The Carrington Event

1. CME speed is 2.37 × 10⁶ m/s or 8.5 million km/h

Media Attributions

• “Heliocentric model from Nicolaus Copernicus’ De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres)” by Copernicus is in the public domain.
• “January 11, 2014 NOAA jet stream wind speed map” is in the public domain.
• “Jupiter Eye to Io” by NASA is in the public domain.