General Physics (1) PHYS101 PDF

General physics (1) For the first level students Lecturer: Dr. Yasmeen Ali Taya Physics department, science faculty Sohag University First term 2025  c h a p t e r 1 Physics...

General physics (1) For the first level students Lecturer: Dr. Yasmeen Ali Taya Physics department, science faculty Sohag University First term 2025  c h a p t e r 1 Physics and Measurement 1.1 Standards of Length, Mass, and Time 1.2 Matter and Model Building 1.3 Dimensional Analysis 1.4 Conversion of Units 1.5 Estimates and Order-of- Magnitude Calculations 1.6 Significant Figures Stonehenge, in southern England, Like all other sciences, physics is based on experimental observations and quantitative was built thousands of years ago. measurements. The main objectives of physics are to identify a limited number of funda- Various theories have been proposed mental laws that govern natural phenomena and use them to develop theories that can pre- about its function, including a dict the results of future experiments. The fundamental laws used in developing theories are burial ground, a healing site, and a place for ancestor worship. One expressed in the language of mathematics, the tool that provides a bridge between theory of the more intriguing theories and experiment. suggests that Stonehenge was an When there is a discrepancy between the prediction of a theory and experimental observatory, allowing measurements results, new or modified theories must be formulated to remove the discrepancy. Many of some of the quantities discussed times a theory is satisfactory only under limited conditions; a more general theory might be in this chapter, such as position of satisfactory without such limitations. For example, the laws of motion discovered by Isaac objects in space and time intervals between repeating celestial events. Newton (1642–1727) accurately describe the motion of objects moving at normal speeds but (Stephen Inglis/Shutterstock.com) do not apply to objects moving at speeds comparable to the speed of light. In contrast, the special theory of relativity developed later by Albert Einstein (1879–1955) gives the same results as Newton’s laws at low speeds but also correctly describes the motion of objects at speeds approaching the speed of light. Hence, Einstein’s special theory of relativity is a more Interactive content general theory of motion than that formed from Newton’s laws. from this and other chapters may be assigned online in Enhanced Classical physics includes the principles of classical mechanics, thermodynamics, optics, WebAssign. and electromagnetism developed before 1900. Important contributions to classical physics 2  1.1 Standards of Length, Mass, and Time 3 were provided by Newton, who was also one of the originators of calculus as a mathemati- cal tool. Major developments in mechanics continued in the 18th century, but the fields of thermodynamics and electromagnetism were not developed until the latter part of the 19th century, principally because before that time the apparatus for controlled experiments in these disciplines was either too crude or unavailable. A major revolution in physics, usually referred to as modern physics, began near the end of the 19th century. Modern physics developed mainly because many physical phenomena could not be explained by classical physics. The two most important developments in this modern era were the theories of relativity and quantum mechanics. Einstein’s special the- ory of relativity not only correctly describes the motion of objects moving at speeds com- parable to the speed of light; it also completely modifies the traditional concepts of space, time, and energy. The theory also shows that the speed of light is the upper limit of the speed of an object and that mass and energy are related. Quantum mechanics was formu- lated by a number of distinguished scientists to provide descriptions of physical phenomena at the atomic level. Many practical devices have been developed using the principles of quantum mechanics. Scientists continually work at improving our understanding of fundamental laws. Numerous technological advances in recent times are the result of the efforts of many scientists, engineers, and technicians, such as unmanned planetary explorations, a vari- ety of developments and potential applications in nanotechnology, microcircuitry and high-speed computers, sophisticated imaging techniques used in scientific research and medicine, and several remarkable results in genetic engineering. The effects of such devel- opments and discoveries on our society have indeed been great, and it is very likely that future discoveries and developments will be exciting, challenging, and of great benefit to humanity. 1.1 Standards of Length, Mass, and Time To describe natural phenomena, we must make measurements of various aspects of nature. Each measurement is associated with a physical quantity, such as the length of an object. The laws of physics are expressed as mathematical relation- ships among physical quantities that we will introduce and discuss throughout the book. In mechanics, the three fundamental quantities are length, mass, and time. All other quantities in mechanics can be expressed in terms of these three. If we are to report the results of a measurement to someone who wishes to repro- duce this measurement, a standard must be defined. It would be meaningless if a visitor from another planet were to talk to us about a length of 8 “glitches” if we do not know the meaning of the unit glitch. On the other hand, if someone familiar with our system of measurement reports that a wall is 2 meters high and our unit of length is defined to be 1 meter, we know that the height of the wall is twice our basic length unit. Whatever is chosen as a standard must be readily accessible and must possess some property that can be measured reliably. Measurement standards used by different people in different places—throughout the Universe—must yield the same result. In addition, standards used for measurements must not change with time. In 1960, an international committee established a set of standards for the fun- damental quantities of science. It is called the SI (Système International), and its fundamental units of length, mass, and time are the meter, kilogram, and second, respectively. Other standards for SI fundamental units established by the commit- tee are those for temperature (the kelvin), electric current (the ampere), luminous intensity (the candela), and the amount of substance (the mole). 4 Chapter 1 Physics and Measurement Length We can identify length as the distance between two points in space. In 1120, the king of England decreed that the standard of length in his country would be named the yard and would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm. Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV. Neither of these standards is constant in time; when a new king took the throne, length measure- ments changed! The French standard prevailed until 1799, when the legal standard of length in France became the meter (m), defined as one ten-millionth of the distance from the equator to the North Pole along one particular longitudinal line that passes through Paris. Notice that this value is an Earth-based standard that does not satisfy the requirement that it can be used throughout the Universe. As recently as 1960, the length of the meter was defined as the distance between two lines on a specific platinum–iridium bar stored under controlled conditions in France. Current requirements of science and technology, however, necessitate more accuracy than that with which the separation between the lines on the bar Pitfall Prevention 1.1 can be determined. In the 1960s and 1970s, the meter was defined as 1 650 763.73 Reasonable Values Generating wavelengths1 of orange-red light emitted from a krypton-86 lamp. In October 1983, intuition about typical values of however, the meter was redefined as the distance traveled by light in vacuum dur- quantities when solving problems is important because you must ing a time of 1/299 792 458 second. In effect, this latest definition establishes that think about your end result and the speed of light in vacuum is precisely 299 792 458 meters per second. This defi- determine if it seems reasonable. nition of the meter is valid throughout the Universe based on our assumption that For example, if you are calculating light is the same everywhere. the mass of a housefly and arrive Table 1.1 lists approximate values of some measured lengths. You should study at a value of 100 kg, this answer is unreasonable and there is an error this table as well as the next two tables and begin to generate an intuition for what somewhere. is meant by, for example, a length of 20 centimeters, a mass of 100 kilograms, or a time interval of 3.2 3 107 seconds. Mass The SI fundamental unit of mass, the kilogram (kg), is defined as the mass of a spe- cific platinum–iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France. This mass standard was established in 1887 and Table 1.1 Approximate Values of Some Measured Lengths Length (m) Distance from the Earth to the most remote known quasar 1.4 3 1026 Distance from the Earth to the most remote normal galaxies 9 3 1025 Distance from the Earth to the nearest large galaxy (Andromeda) 2 3 1022 Distance from the Sun to the nearest star (Proxima Centauri) 4 3 1016 One light-year 9.46 3 1015 Mean orbit radius of the Earth about the Sun 1.50 3 1011 Mean distance from the Earth to the Moon 3.84 3 108 Distance from the equator to the North Pole 1.00 3 107 Mean radius of the Earth 6.37 3 106 Typical altitude (above the surface) of a satellite orbiting the Earth 2 3 105 Length of a football field 9.1 3 101 Length of a housefly 5 3 1023 Size of smallest dust particles , 1024 Size of cells of most living organisms , 1025 Diameter of a hydrogen atom , 10210 Diameter of an atomic nucleus , 10214 Diameter of a proton , 10215 1We will use the standard international notation for numbers with more than three digits, in which groups of three digits are separated by spaces rather than commas. Therefore, 10 000 is the same as the common American notation of 10,000. Similarly, p 5 3.14159265 is written as 3.141 592 65.  1.1 Standards of Length, Mass, and Time 5 Table 1.2 Table 1.3 Approximate Values of Approximate Masses of Some Time Intervals Reproduced with permission of the BIPM, which retains full internationally protected copyright. Various Objects Time Interval (s) Mass (kg) Age of the Universe 4 3 1017 Observable Age of the Earth 1.3 3 1017 Universe , 1052 Average age of a college student 6.3 3 108 Milky Way One year 3.2 3 107 galaxy , 1042 One day 8.6 3 104 Sun 1.99 3 1030 One class period 3.0 3 103 Earth 5.98 3 1024 Time interval between normal Moon 7.36 3 1022 heartbeats 8 3 1021 Shark , 103 Period of audible sound waves , 1023 a Human , 102 Period of typical radio waves , 1026 Frog , 1021 Period of vibration of an atom Mosquito , 1025 in a solid , 10213 Bacterium , 1 3 10215 Period of visible light waves , 10215 Hydrogen atom 1.67 3 10227 Duration of a nuclear collision , 10222 Electron 9.11 3 10231 Time interval for light to cross a proton , 10224 has not been changed since that time because platinum–iridium is an unusually stable alloy. A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland (Fig. 1.1a). Table 1.2 lists approximate values of the masses of various objects. Time Before 1967, the standard of time was defined in terms of the mean solar day. (A solar day is the time interval between successive appearances of the Sun at the highest point AP Photo/Focke Strangmann it reaches in the sky each day.) The fundamental unit of a second (s) was defined as 1 1 1 1 60 2 1 60 2 1 24 2 of a mean solar day. This definition is based on the rotation of one planet, the Earth. Therefore, this motion does not provide a time standard that is universal. In 1967, the second was redefined to take advantage of the high precision attain- able in a device known as an atomic clock (Fig. 1.1b), which measures vibrations of cesium atoms. One second is now defined as 9 192 631 770 times the period of b vibration of radiation from the cesium-133 atom.2 Approximate values of time intervals are presented in Table 1.3. Figure 1.1 (a) The National In addition to SI, another system of units, the U.S. customary system, is still used in Standard Kilogram No. 20, an accurate copy of the International the United States despite acceptance of SI by the rest of the world. In this system, Standard Kilogram kept at Sèvres, the units of length, mass, and time are the foot (ft), slug, and second, respectively. France, is housed under a double In this book, we shall use SI units because they are almost universally accepted in bell jar in a vault at the National science and industry. We shall make some limited use of U.S. customary units in Institute of Standards and Tech- the study of classical mechanics. nology. (b) A cesium fountain atomic clock. The clock will nei- In addition to the fundamental SI units of meter, kilogram, and second, we can ther gain nor lose a second in 20 also use other units, such as millimeters and nanoseconds, where the prefixes milli- million years. and nano- denote multipliers of the basic units based on various powers of ten. Prefixes for the various powers of ten and their abbreviations are listed in Table 1.4 (page 6). For example, 1023 m is equivalent to 1 millimeter (mm), and 103 m corre- sponds to 1 kilometer (km). Likewise, 1 kilogram (kg) is 103 grams (g), and 1 mega volt (MV) is 106 volts (V). The variables length, time, and mass are examples of fundamental quantities. Most other variables are derived quantities, those that can be expressed as a mathematical combination of fundamental quantities. Common examples are area (a product of two lengths) and speed (a ratio of a length to a time interval). 2Period is defined as the time interval needed for one complete vibration. 6 Chapter 1 Physics and Measurement Table 1.4 Prefixes for Powers of Ten Power Prefix Abbreviation Power Prefix Abbreviation 10224 yocto y 103   kilo k 10221 zepto z 106   mega M 10218 atto a 109   giga G 10215 femto f 1012 tera T 10212 pico p 1015 peta P 1029   nano n 1018 exa E 1026   micro m 1021 zetta Z 1023   milli m 1024 yotta Y 1022   centi c 1021   deci d A table of the letters in the  Another example of a derived quantity is density. The density r (Greek letter Greek alphabet is provided rho) of any substance is defined as its mass per unit volume: on the back endpaper of this book. m r; (1.1) V In terms of fundamental quantities, density is a ratio of a mass to a product of three lengths. Aluminum, for example, has a density of 2.70 3 103 kg/m3, and iron has a density of 7.86 3 103 kg/m3. An extreme difference in density can be imagined by thinking about holding a 10-centimeter (cm) cube of Styrofoam in one hand and a 10-cm cube of lead in the other. See Table 14.1 in Chapter 14 for densities of several materials. Don Farrall/Photodisc/ Q uick Quiz 1.1 In a machine shop, two cams are produced, one of aluminum Getty Images and one of iron. Both cams have the same mass. Which cam is larger? (a) The aluminum cam is larger. (b) The iron cam is larger. (c) Both cams have the same size. A piece of gold consists of gold atoms. 1.2 Matter and Model Building If physicists cannot interact with some phenomenon directly, they often imagine At the center a model for a physical system that is related to the phenomenon. For example, we of each atom cannot interact directly with atoms because they are too small. Therefore, we build is a nucleus. a mental model of an atom based on a system of a nucleus and one or more elec- trons outside the nucleus. Once we have identified the physical components of the model, we make predictions about its behavior based on the interactions among Inside the nucleus are the components of the system or the interaction between the system and the envi- protons ronment outside the system. (orange) and As an example, consider the behavior of matter. A sample of solid gold is shown neutrons at the top of Figure 1.2. Is this sample nothing but wall-to-wall gold, with no empty (gray). space? If the sample is cut in half, the two pieces still retain their chemical iden- tity as solid gold. What if the pieces are cut again and again, indefinitely? Will the Protons and smaller and smaller pieces always be gold? Such questions can be traced to early neutrons are Greek philosophers. Two of them—Leucippus and his student Democritus—could composed of p not accept the idea that such cuttings could go on forever. They developed a model quarks. The for matter by speculating that the process ultimately must end when it produces a quark u u particle that can no longer be cut. In Greek, atomos means “not sliceable.” From this composition of a proton is Greek term comes our English word atom. d shown here. The Greek model of the structure of matter was that all ordinary matter consists of atoms, as suggested in the middle of Figure 1.2. Beyond that, no additional struc- Figure 1.2 Levels of organization ture was specified in the model; atoms acted as small particles that interacted with in matter. one another, but internal structure of the atom was not a part of the model.  1.3 Dimensional Analysis 7 In 1897, J. J. Thomson identified the electron as a charged particle and as a constituent of the atom. This led to the first atomic model that contained internal structure. We shall discuss this model in Chapter 42. Following the discovery of the nucleus in 1911, an atomic model was developed in which each atom is made up of electrons surrounding a central nucleus. A nucleus of gold is shown in Figure 1.2. This model leads, however, to a new question: Does the nucleus have structure? That is, is the nucleus a single particle or a collection of particles? By the early 1930s, a model evolved that described two basic entities in the nucleus: protons and neutrons. The proton carries a positive electric charge, and a specific chemical element is identified by the number of protons in its nucleus. This number is called the atomic number of the element. For instance, the nucleus of a hydrogen atom contains one proton (so the atomic number of hydrogen is 1), the nucleus of a helium atom contains two protons (atomic number 2), and the nucleus of a uranium atom contains 92 protons (atomic number 92). In addition to atomic number, a second number—mass number, defined as the number of protons plus neutrons in a nucleus—characterizes atoms. The atomic number of a specific ele- ment never varies (i.e., the number of protons does not vary), but the mass number can vary (i.e., the number of neutrons varies). Is that, however, where the process of breaking down stops? Protons, neutrons, and a host of other exotic particles are now known to be composed of six different varieties of particles called quarks, which have been given the names of up, down, strange, charmed, bottom, and top. The up, charmed, and top quarks have electric charges of 123 that of the proton, whereas the down, strange, and bottom quarks have charges of 213 that of the proton. The proton consists of two up quarks and one down quark as shown at the bottom of Figure 1.2 and labeled u and d. This structure predicts the correct charge for the proton. Likewise, the neutron consists of two down quarks and one up quark, giving a net charge of zero. You should develop a process of building models as you study physics. In this study, you will be challenged with many mathematical problems to solve. One of the most important problem-solving techniques is to build a model for the prob- lem: identify a system of physical components for the problem and make predic- tions of the behavior of the system based on the interactions among its components or the interaction between the system and its surrounding environment. 1.3 Dimensional Analysis In physics, the word dimension denotes the physical nature of a quantity. The dis- tance between two points, for example, can be measured in feet, meters, or fur- longs, which are all different ways of expressing the dimension of length. The symbols we use in this book to specify the dimensions of length, mass, and time are L, M, and T, respectively.3 We shall often use brackets [ ] to denote the dimensions of a physical quantity. For example, the symbol we use for speed in this book is v, and in our notation, the dimensions of speed are written [v] 5 L/T. As another example, the dimensions of area A are [A] 5 L2. The dimensions and units of area, volume, speed, and acceleration are listed in Table 1.5. The dimensions of other quantities, such as force and energy, will be described as they are introduced in the text. Table 1.5 Dimensions and Units of Four Derived Quantities Quantity Area (A) Volume (V ) Speed (v) Acceleration (a) Dimensions L2 L3 L/T L/T2 SI units m2 m3 m/s m/s2 U.S. customary units ft 2 ft 3 ft/s ft/s2 3The dimensions of a quantity will be symbolized by a capitalized, nonitalic letter such as L or T. The algebraic symbol for the quantity itself will be an italicized letter such as L for the length of an object or t for time. 8 Chapter 1 Physics and Measurement In many situations, you may have to check a specific equation to see if it matches your expectations. A useful procedure for doing that, called dimensional analysis, can be used because dimensions can be treated as algebraic quantities. For exam- ple, quantities can be added or subtracted only if they have the same dimensions. Furthermore, the terms on both sides of an equation must have the same dimen- sions. By following these simple rules, you can use dimensional analysis to deter- mine whether an expression has the correct form. Any relationship can be correct only if the dimensions on both sides of the equation are the same. To illustrate this procedure, suppose you are interested in an equation for the Pitfall Prevention 1.2 position x of a car at a time t if the car starts from rest at x 5 0 and moves with con- Symbols for Quantities Some stant acceleration a. The correct expression for this situation is x 5 12 at 2 as we show quantities have a small number of symbols that represent them. in Chapter 2. The quantity x on the left side has the dimension of length. For the For example, the symbol for time equation to be dimensionally correct, the quantity on the right side must also have is almost always t. Other quanti- the dimension of length. We can perform a dimensional check by substituting the ties might have various symbols dimensions for acceleration, L/T2 (Table 1.5), and time, T, into the equation. That depending on the usage. Length is, the dimensional form of the equation x 5 12 at 2 is may be described with symbols such as x, y, and z (for position); r (for radius); a, b, and c (for the L # 2 L5 T 5L legs of a right triangle); , (for the T2 length of an object); d (for a dis- tance); h (for a height); and The dimensions of time cancel as shown, leaving the dimension of length on the so forth. right-hand side to match that on the left. A more general procedure using dimensional analysis is to set up an expression of the form x ~ an t m where n and m are exponents that must be determined and the symbol ~ indicates a proportionality. This relationship is correct only if the dimensions of both sides are the same. Because the dimension of the left side is length, the dimension of the right side must also be length. That is, 3 ant m 4 5 L 5 L1T0 Because the dimensions of acceleration are L/T2 and the dimension of time is T, we have 1 L/T2 2 n Tm 5 L1T0 S 1 Ln Tm22n 2 5 L1T0 The exponents of L and T must be the same on both sides of the equation. From the exponents of L, we see immediately that n 5 1. From the exponents of T, we see that m 2 2n 5 0, which, once we substitute for n, gives us m 5 2. Returning to our original expression x ~ ant m , we conclude that x ~ at 2. Q uick Quiz 1.2 True or False: Dimensional analysis can give you the numerical value of constants of proportionality that may appear in an algebraic expression. Example 1.1    Analysis of an Equation Show that the expression v 5 at, where v represents speed, a acceleration, and t an instant of time, is dimensionally correct. Solution L Identify the dimensions of v from Table 1.5: 3v 4 5 T  1.4 Conversion of Units 9 ▸ 1.1 c o n t i n u e d L L Identify the dimensions of a from Table 1.5 and multiply 3 at 4 5 T 5 by the dimensions of t : T2 T Therefore, v 5 at is dimensionally correct because we have the same dimensions on both sides. (If the expression were given as v 5 at 2, it would be dimensionally incorrect. Try it and see!) Example 1.2    Analysis of a Power Law Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is propor- tional to some power of r, say r n , and some power of v, say v m. Determine the values of n and m and write the simplest form of an equation for the acceleration. Solution Write an expression for a with a dimensionless constant a 5 kr n v m of proportionality k: L L m Ln1m Substitute the dimensions of a, r, and v: 2 5 Ln a b 5 m T T T Equate the exponents of L and T so that the dimen- n 1 m 5 1 and m 5 2 sional equation is balanced: Solve the two equations for n: n 5 21 v2 Write the acceleration expression: a 5 kr21 v 2 5 k r In Section 4.4 on uniform circular motion, we show that k 5 1 if a consistent set of units is used. The constant k would not equal 1 if, for example, v were in km/h and you wanted a in m/s2. Pitfall Prevention 1.3 1.4 Conversion of Units Always Include Units When per- forming calculations with numeri- Sometimes it is necessary to convert units from one measurement system to another cal values, include the units for or convert within a system (for example, from kilometers to meters). Conversion every quantity and carry the units factors between SI and U.S. customary units of length are as follows: through the entire calculation. Avoid the temptation to drop the 1 mile 5 1 609 m 5 1.609 km 1 ft 5 0.304 8 m 5 30.48 cm units early and then attach the 1 m 5 39.37 in. 5 3.281 ft 1 in. 5 0.025 4 m 5 2.54 cm (exactly) expected units once you have an answer. By including the units in A more complete list of conversion factors can be found in Appendix A. every step, you can detect errors if Like dimensions, units can be treated as algebraic quantities that can can- the units for the answer turn out cel each other. For example, suppose we wish to convert 15.0 in. to centimeters. to be incorrect. Because 1 in. is defined as exactly 2.54 cm, we find that 2.54 cm 15.0 in. 5 1 15.0 in. 2 a b 5 38.1 cm 1 in. where the ratio in parentheses is equal to 1. We express 1 as 2.54 cm/1 in. (rather than 1 in./2.54 cm) so that the unit “inch” in the denominator cancels with the unit in the original quantity. The remaining unit is the centimeter, our desired result. 10 Chapter 1 Physics and Measurement Q uick Quiz 1.3 The distance between two cities is 100 mi. What is the number of kilometers between the two cities? (a) smaller than 100 (b) larger than 100 (c) equal to 100 Example 1.3    Is He Speeding? On an interstate highway in a rural region of Wyoming, a car is traveling at a speed of 38.0 m/s. Is the driver exceeding the speed limit of 75.0 mi/h? Solution 1 mi Convert meters in the speed to miles: 1 38.0 m/s 2 a b 5 2.36 3 1022 mi/s 1 609 m 60 s 60 min Convert seconds to hours: 1 2.36 3 1022 mi/s 2 a b a b 5 85.0 mi/h 1 min 1h The driver is indeed exceeding the speed limit and should slow down. W h at If ? What if the driver were from outside the United States and is familiar with speeds measured in kilometers per hour? What is the speed of the car in km/h? Answer We can convert our final answer to the appropriate units: 1.609 km 1 85.0 mi/h 2 a b 5 137 km/h 1 mi © Cengage Learning/Ed Dodd Figure 1.3 shows an automobile speedometer displaying speeds in both mi/h and km/h. Can you check the conversion we just performed using this photograph? Figure 1.3 The speedometer of a vehicle that shows speeds in both miles per hour and kilometers per hour. 1.5 Estimates and Order-of-Magnitude Calculations Suppose someone asks you the number of bits of data on a typical musical com- pact disc. In response, it is not generally expected that you would provide the exact number but rather an estimate, which may be expressed in scientific notation. The estimate may be made even more approximate by expressing it as an order of magni- tude, which is a power of ten determined as follows: 1. Express the number in scientific notation, with the multiplier of the power of ten between 1 and 10 and a unit. 2. If the multiplier is less than 3.162 (the square root of 10), the order of mag- nitude of the number is the power of 10 in the scientific notation. If the multiplier is greater than 3.162, the order of magnitude is one larger than the power of 10 in the scientific notation. We use the symbol , for “is on the order of.” Use the procedure above to verify the orders of magnitude for the following lengths: 0.008 6 m , 1022 m   0.002 1 m , 1023 m   720 m , 103 m  1.6 Significant Figures 11 Usually, when an order-of-magnitude estimate is made, the results are reliable to within about a factor of 10. If a quantity increases in value by three orders of magni- tude, its value increases by a factor of about 103 5 1 000. Inaccuracies caused by guessing too low for one number are often canceled by other guesses that are too high. You will find that with practice your guessti- mates become better and better. Estimation problems can be fun to work because you freely drop digits, venture reasonable approximations for unknown numbers, make simplifying assumptions, and turn the question around into something you can answer in your head or with minimal mathematical manipulation on paper. Because of the simplicity of these types of calculations, they can be performed on a small scrap of paper and are often called “back-of-the-envelope calculations.” Example 1.4    Breaths in a Lifetime Estimate the number of breaths taken during an average human lifetime. Solution We start by guessing that the typical human lifetime is about 70 years. Think about the average number of breaths that a person takes in 1 min. This number varies depending on whether the person is exercising, sleeping, angry, serene, and so forth. To the nearest order of magnitude, we shall choose 10 breaths per minute as our estimate. (This estimate is certainly closer to the true average value than an estimate of 1 breath per minute or 100 breaths per minute.) 400 days 25 h 60 min Find the approximate number of minutes in a year: 1 yr a b a b a b 5 6 3 105 min 1 yr 1 day 1h Find the approximate number of minutes in a 70-year number of minutes 5 (70 yr)(6 3 105 min/yr) lifetime: 5 4 3 107 min Find the approximate number of breaths in a lifetime: number of breaths 5 (10 breaths/min)(4 3 107 min) 5 4 3 108 breaths Therefore, a person takes on the order of 109 breaths in a lifetime. Notice how much simpler it is in the first calculation above to multiply 400 3 25 than it is to work with the more accurate 365 3 24. W h at If ? What if the average lifetime were estimated as 80 years instead of 70? Would that change our final estimate? Answer We could claim that (80 yr)(6 3 105 min/yr) 5 5 3 107 min, so our final estimate should be 5 3 108 breaths. This answer is still on the order of 109 breaths, so an order-of-magnitude estimate would be unchanged. 1.6 Significant Figures When certain quantities are measured, the measured values are known only to within the limits of the experimental uncertainty. The value of this uncertainty can depend on various factors, such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed. The number of significant figures in a measurement can be used to express something about the uncertainty. The number of significant figures is related to the number of numeri- cal digits used to express the measurement, as we discuss below. As an example of significant figures, suppose we are asked to measure the radius of a compact disc using a meterstick as a measuring instrument. Let us assume the accuracy to which we can measure the radius of the disc is 60.1 cm. Because of the uncertainty of 60.1 cm, if the radius is measured to be 6.0 cm, we can claim only that its radius lies somewhere between 5.9 cm and 6.1 cm. In this case, we say that the measured value of 6.0 cm has two significant figures. Note that the 12 Chapter 1 Physics and Measurement significant figures include the first estimated digit. Therefore, we could write the radius as (6.0 6 0.1) cm. Zeros may or may not be significant figures. Those used to position the decimal point in such numbers as 0.03 and 0.007 5 are not significant. Therefore, there are one and two significant figures, respectively, in these two values. When the zeros come after other digits, however, there is the possibility of misinterpretation. For example, suppose the mass of an object is given as 1 500 g. This value is ambigu- ous because we do not know whether the last two zeros are being used to locate the decimal point or whether they represent significant figures in the measurement. To remove this ambiguity, it is common to use scientific notation to indicate the number of significant figures. In this case, we would express the mass as 1.5 3 103 g if there are two significant figures in the measured value, 1.50 3 103 g if there are three sig- nificant figures, and 1.500 3 103 g if there are four. The same rule holds for numbers less than 1, so 2.3 3 1024 has two significant figures (and therefore could be written 0.000 23) and 2.30 3 1024 has three significant figures (also written as 0.000 230). In problem solving, we often combine quantities mathematically through mul- tiplication, division, addition, subtraction, and so forth. When doing so, you must make sure that the result has the appropriate number of significant figures. A good rule of thumb to use in determining the number of significant figures that can be claimed in a multiplication or a division is as follows: When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the quantity hav- ing the smallest number of significant figures. The same rule applies to division. Let’s apply this rule to find the area of the compact disc whose radius we mea- sured above. Using the equation for the area of a circle, A 5 pr 2 5 p 1 6.0 cm 2 2 5 1.1 3 102 cm2 If you perform this calculation on your calculator, you will likely see 113.097 335 5. It should be clear that you don’t want to keep all of these digits, but you might be tempted to report the result as 113 cm2. This result is not justified because it has three significant figures, whereas the radius only has two. Therefore, we must report the result with only two significant figures as shown above. For addition and subtraction, you must consider the number of decimal places Pitfall Prevention 1.4 when you are determining how many significant figures to report: Read Carefully Notice that the rule for addition and subtraction When numbers are added or subtracted, the number of decimal places in the is different from that for multipli- result should equal the smallest number of decimal places of any term in the cation and division. For addition and subtraction, the important sum or difference. consideration is the number of decimal places, not the number of As an example of this rule, consider the sum significant figures. 23.2 1 5.174 5 28.4 Notice that we do not report the answer as 28.374 because the lowest number of decimal places is one, for 23.2. Therefore, our answer must have only one decimal place. The rule for addition and subtraction can often result in answers that have a dif- ferent number of significant figures than the quantities with which you start. For example, consider these operations that satisfy the rule: 1.000 1 1 0.000 3 5 1.000 4 1.002 2 0.998 5 0.004 In the first example, the result has five significant figures even though one of the terms, 0.000 3, has only one significant figure. Similarly, in the second calcu- lation, the result has only one significant figure even though the numbers being subtracted have four and three, respectively.  Summary 13 In this book, most of the numerical examples and end-of-chapter problems Significant figure guidelines WW will yield answers having three significant figures. When carrying out estima- used in this book tion calculations, we shall typically work with a single significant figure. If the number of significant figures in the result of a calculation must be reduced, Pitfall Prevention 1.5 there is a general rule for rounding numbers: the last digit retained is increased by Symbolic Solutions When solving 1 if the last digit dropped is greater than 5. (For example, 1.346 becomes 1.35.) problems, it is very useful to per- If the last digit dropped is less than 5, the last digit retained remains as it is. (For form the solution completely in example, 1.343 becomes 1.34.) If the last digit dropped is equal to 5, the remaining algebraic form and wait until the digit should be rounded to the nearest even number. (This rule helps avoid accu- very end to enter numerical values mulation of errors in long arithmetic processes.) into the final symbolic expres- sion. This method will save many A technique for avoiding error accumulation is to delay the rounding of num- calculator keystrokes, especially if bers in a long calculation until you have the final result. Wait until you are ready to some quantities cancel so that you copy the final answer from your calculator before rounding to the correct number never have to enter their values of significant figures. In this book, we display numerical values rounded off to two into your calculator! In addition, or three significant figures. This occasionally makes some mathematical manipula- you will only need to round once, on the final result. tions look odd or incorrect. For instance, looking ahead to Example 3.5 on page 69, you will see the operation 217.7 km 1 34.6 km 5 17.0 km. This looks like an incor- rect subtraction, but that is only because we have rounded the numbers 17.7 km and 34.6 km for display. If all digits in these two intermediate numbers are retained and the rounding is only performed on the final number, the correct three-digit result of 17.0 km is obtained. Example 1.5    Installing a Carpet A carpet is to be installed in a rectangular room whose length is measured to be 12.71 m and whose width is measured to be 3.46 m. Find the area of the room. Solution If you multiply 12.71 m by 3.46 m on your calculator, you will see an answer of 43.976 6 m2. How many of these num- bers should you claim? Our rule of thumb for multiplication tells us that you can claim only the number of significant figures in your answer as are present in the measured quantity having the lowest number of significant figures. In this example, the lowest number of significant figures is three in 3.46 m, so we should express our final answer as 44.0 m2. Summary Definitions The three fundamental physical quantities of The density of a substance is defined as its mass per mechanics are length, mass, and time, which in the SI unit volume: system have the units meter (m), kilogram (kg), and m r; (1.1) second (s), respectively. These fundamental quantities V cannot be defined in terms of more basic quantities. continued 14 Chapter 1 Physics and Measurement Concepts and Principles The method of dimensional analysis is very power- When you compute a result from several measured ful in solving physics problems. Dimensions can be numbers, each of which has a certain accuracy, you treated as algebraic quantities. By making estimates should give the result with the correct number of sig- and performing order-of-magnitude calculations, you nificant figures. should be able to approximate the answer to a prob- When multiplying several quantities, the number of lem when there is not enough information available to significant ­f igures in the final answer is the same as the specify an exact solution completely. number of significant figures in the quantity having the smallest number of significant figures. The same rule applies to division. When numbers are added or subtracted, the number of decimal places in the result should equal the small- est number of decimal places of any term in the sum or difference. Objective Questions 1. denotes answer available in Student Solutions Manual/Study Guide 1. One student uses a meterstick to measure the thick- give them equal rank in your list. (a) 0.032 kg (b) 15 g ness of a textbook and obtains 4.3 cm 6 0.1 cm. Other (c) 2.7 3 105 mg (d) 4.1 3 1028 Gg (e) 2.7 3 108 mg students measure the thickness with vernier calipers 6. What is the sum of the measured values 21.4 s 1 15 s 1 and obtain four different measurements: (a) 4.32 cm 17.17 s 1 4.00 3 s? (a) 57.573 s (b) 57.57 s (c) 57.6 s 6 0.01 cm, (b) 4.31 cm 6 0.01 cm, (c) 4.24 cm 6 0.01 cm, (d) 58 s (e) 60 s and (d) 4.43 cm 6 0.01 cm. Which of these four mea- surements, if any, agree with that obtained by the first 7. Which of the following is the best estimate for the mass student? of all the people living on the Earth? (a) 2 3 108 kg (b) 1 3 109 kg (c) 2 3 1010 kg (d) 3 3 1011 kg 2. A house is advertised as having 1 420 square feet under (e) 4 3 1012 kg its roof. What is its area in square meters? (a) 4 660 m2 (b) 432 m2 (c) 158 m2 (d) 132 m2 (e) 40.2 m2 8. (a) If an equation is dimensionally correct, does that mean that the equation must be true? (b) If an equa- 3. Answer each question yes or no. Must two quantities tion is not dimensionally correct, does that mean that have the same dimensions (a) if you are adding them? the equation cannot be true? (b) If you are multiplying them? (c) If you are subtract- ing them? (d) If you are dividing them? (e) If you are 9. Newton’s second law of motion (Chapter 5) says that the equating them? mass of an object times its acceleration is equal to the net force on the object. Which of the following gives 4. The price of gasoline at a particular station is 1.5 euros the correct units for force? (a) kg ? m/s2 (b) kg ? m2/s2 per liter. An American student can use 33 euros to buy (c) kg/m ? s2 (d) kg ? m2/s (e) none of those answers gasoline. Knowing that 4 quarts make a gallon and that 1 liter is close to 1 quart, she quickly reasons that she 10. A calculator displays a result as 1.365 248 0 3 107 kg. can buy how many gallons of gasoline? (a) less than The estimated uncertainty in the result is 62%. How 1 gallon (b) about 5 gallons (c) about 8 gallons (d) more many digits should be included as significant when the than 10 gallons result is written down? (a) zero (b) one (c) two (d) three (e) four 5. Rank the following five quantities in order from the largest to the smallest. If two of the quantities are equal, Conceptual Questions 1. denotes answer available in Student Solutions Manual/Study Guide 1. Suppose the three fundamental standards of the met- 2. Why is the metric system of units considered superior ric system were length, density, and time rather than to most other systems of units? length, mass, and time. The standard of density in this 3. What natural phenomena could serve as alternative system is to be defined as that of water. What consid- time standards? erations about water would you need to address to make sure that the standard of density is as accurate as 4. Express the following quantities using the prefixes given possible? in Table 1.4. (a) 3 3 1024 m (b) 5 3 1025 s (c) 72 3 102 g  c h a p t e r Motion in One Dimension 2 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed 2.3 Analysis Model: Particle Under Constant Velocity 2.4 Acceleration 2.5 Motion Diagrams 2.6 Analysis Model: Particle Under Constant Acceleration 2.7 Freely Falling Objects 2.8 Kinematic Equations Derived from Calculus General Problem-Solving Strategy As a first step in studying classical mechanics, we describe the motion of an object In drag racing, a driver wants as while ignoring the interactions with external agents that might be affecting or modifying large an acceleration as possible. In a distance of one-quarter mile, that motion. This portion of classical mechanics is called kinematics. (The word kinematics a vehicle reaches speeds of more has the same root as cinema.) In this chapter, we consider only motion in one dimension, than 320 mi/h, covering the entire that is, motion of an object along a straight line. distance in under 5 s. (George Lepp/ From everyday experience, we recognize that motion of an object represents a continu- Stone/Getty Images) ous change in the object’s position. In physics, we can categorize motion into three types: translational, rotational, and vibrational. A car traveling on a highway is an example of translational motion, the Earth’s spin on its axis is an example of rotational motion, and the back-and-forth movement of a pendulum is an example of vibrational motion. In this and the next few chapters, we are concerned only with translational motion. (Later in the book we shall discuss rotational and vibrational motions.) In our study of translational motion, we use what is called the particle model and describe the moving object as a particle regardless of its size. Remember our discussion of making models for physical situations in Section 1.2. In general, a particle is a point-like object, that is, an object that has mass but is of infinitesimal size. For example, if we wish to describe the motion of the Earth around the Sun, we can treat the Earth as a particle and 21 22 Chapter 2 Motion in One Dimension obtain reasonably accurate data about its orbit. This approximation is justified because the radius of the Earth’s orbit is large compared with the dimensions of the Earth and the Sun. As an example on a much smaller scale, it is possible to explain the pressure exerted by a gas on the walls of a container by treating the gas molecules as particles, without regard for the internal structure of the molecules. 2.1 Position, Velocity, and Speed Position  A particle’s position x is the location of the particle with respect to a chosen ref- erence point that we can consider to be the origin of a coordinate system. The motion of a particle is completely known if the particle’s position in space is known at all times. Consider a car moving back and forth along the x axis as in Figure 2.1a. When we begin collecting position data, the car is 30 m to the right of the reference posi- tion x 5 0. We will use the particle model by identifying some point on the car, perhaps the front door handle, as a particle representing the entire car. We start our clock, and once every 10 s we note the car’s position. As you can see from Table 2.1, the car moves to the right (which we have defined as the positive direction) during the first 10 s of motion, from position A to position B. After B, the position values begin to decrease, suggesting the car is backing up from position B through position F. In fact, at D, 30 s after we start measuring, the car is at the Table 2.1 Position of origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than the Car at Various Times 50 m to the left of x 5 0 when we stop recording information after our sixth data Position t (s) x (m) point. A graphical representation of this information is presented in Figure 2.1b. A  0 30 Such a plot is called a position–time graph. B 10 52 Notice the alternative representations of information that we have used for the C 20 38 motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a D 30 0 graphical representation. Table 2.1 is a tabular representation of the same information. E 40 237 Using an alternative representation is often an excellent strategy for understanding F 50 253 the situation in a given problem. The ultimate goal in many problems is a math- The car moves to the right between positions A and B. A B x (m) 60 B x (m) x C 40 60 50 40 30 20 10 0 10 20 30 40 50 60 A 20 t F E D C 0 D x (m) 60 50 40 30 20 10 0 10 20 30 40 50 60 20 The car moves to E 40 the left between F positions C and F. 60 t (s) 0 10 20 30 40 50 a b Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car’s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position–time graph) of the motion of the car.  2.1 Position, Velocity, and Speed 23 ematical representation, which can be analyzed to solve for some requested piece of information. Given the data in Table 2.1, we can easily determine the change in position of the car for various time intervals. The displacement Dx of a particle is defined as its change in position in some time interval. As the particle moves from an initial position xi to a final position xf , its displacement is given by Dx ; xf 2 xi (2.1) Displacement WW We use the capital Greek letter delta (D) to denote the change in a quantity. From this definition, we see that Dx is positive if xf is greater than xi and negative if xf is less than xi. Brian Drake/Time Life Pictures/Getty Images It is very important to recognize the difference between displacement and dis- tance traveled. Distance is the length of a path followed by a particle. Consider, for example, the basketball players in Figure 2.2. If a player runs from his own team’s basket down the court to the other team’s basket and then returns to his own bas- ket, the displacement of the player during this time interval is zero because he ended up at the same point as he started: xf 5 xi , so Dx 5 0. During this time interval, however, he moved through a distance of twice the length of the basketball court. Distance is always represented as a positive number, whereas displacement can be either positive or negative. Figure 2.2 On this basketball Displacement is an example of a vector quantity. Many other physical quantities, court, players run back and forth including position, velocity, and acceleration, also are vectors. In general, a vector for the entire game. The distance that the players run over the quantity requires the specification of both direction and magnitude. By contrast, a duration of the game is nonzero. scalar quantity has a numerical value and no direction. In this chapter, we use posi- The displacement of the players tive (1) and negative (2) signs to indicate vector direction. For example, for hori- over the duration of the game is zontal motion let us arbitrarily specify to the right as being the positive direction. approximately zero because they It follows that any object always moving to the right undergoes a positive displace- keep returning to the same point over and over again. ment Dx. 0, and any object moving to the left undergoes a negative displacement so that Dx , 0. We shall treat vector quantities in greater detail in Chapter 3. One very important point has not yet been mentioned. Notice that the data in Table 2.1 result only in the six data points in the graph in Figure 2.1b. Therefore, the motion of the particle is not completely known because we don’t know its posi- tion at all times. The smooth curve drawn through the six points in the graph is only a possibility of the actual motion of the car. We only have information about six instants of time; we have no idea what happened between the data points. The smooth curve is a guess as to what happened, but keep in mind that it is only a guess. If the smooth curve does represent the actual motion of the car, the graph contains complete information about the entire 50-s interval during which we watch the car move. It is much easier to see changes in position from the graph than from a verbal description or even a table of numbers. For example, it is clear that the car covers more ground during the middle of the 50-s interval than at the end. Between posi- tions C and D, the car travels almost 40 m, but during the last 10 s, between posi- tions E and F, it moves less than half that far. A common way of comparing these different motions is to divide the displacement Dx that occurs between two clock readings by the value of that particular time interval Dt. The result turns out to be a very useful ratio, one that we shall use many times. This ratio has been given a special name: the average velocity. The average velocity vx,avg of a particle is defined as the particle’s displacement Dx divided by the time interval Dt during which that displacement occurs: Dx v x,avg ; (2.2) Average velocity WW Dt where the subscript x indicates motion along the x axis. From this definition we see that average velocity has dimensions of length divided by time (L/T), or meters per second in SI units. 24 Chapter 2 Motion in One Dimension The average velocity of a particle moving in one dimension can be positive or negative, depending on the sign of the displacement. (The time interval Dt is always positive.) If the coordinate of the particle increases in time (that is, if xf. xi ), Dx is positive and vx,avg 5 Dx/Dt is positive. This case corresponds to a particle mov- ing in the positive x direction, that is, toward larger values of x. If the coordinate decreases in time (that is, if xf , xi ), Dx is negative and hence vx,avg is negative. This case corresponds to a particle moving in the negative x direction. We can interpret average velocity geometrically by drawing a straight line between any two points on the position–time graph in Figure 2.1b. This line forms the hypotenuse of a right triangle of height Dx and base Dt. The slope of this line is the ratio Dx/Dt, which is what we have defined as average velocity in Equation 2.2. For example, the line between positions A and B in Figure 2.1b has a slope equal to the average velocity of the car between those two times, (52 m 2 30 m)/(10 s 2 0) 5 2.2 m/s. In everyday usage, the terms speed and velocity are interchangeable. In physics, however, there is a clear distinction between these two quantities. Consider a mara- thon runner who runs a distance d of more than 40 km and yet ends up at her starting point. Her total displacement is zero, so her average velocity is zero! None- theless, we need to be able to quantify how fast she was running. A slightly differ- ent ratio accomplishes that for us. The average speed vavg of a particle, a scalar quantity, is defined as the total distance d traveled divided by the total time interval required to travel that distance: d Average speed  v avg ; (2.3) Dt The SI unit of average speed is the same as the unit of average velocity: meters per second. Unlike average velocity, however, average speed has no direction and Pitfall Prevention 2.1 is always expressed as a positive number. Notice the clear distinction between the Average Speed and Average definitions of average velocity and average speed: average velocity (Eq. 2.2) is the Velocity The magnitude of the average velocity is not the aver- displacement divided by the time interval, whereas average speed (Eq. 2.3) is the dis- age speed. For example, consider tance divided by the time interval. the marathon runner discussed Knowledge of the average velocity or average speed of a particle does not provide before Equation 2.3. The mag- information about the details of the trip. For example, suppose it takes you 45.0 s nitude of her average velocity to travel 100 m down a long, straight hallway toward your departure gate at an is zero, but her average speed is clearly not zero. airport. At the 100-m mark, you realize you missed the restroom, and you return back 25.0 m along the same hallway, taking 10.0 s to make the return trip. The magnitude of your average velocity is 175.0 m/55.0 s 5 11.36 m/s. The average speed for your trip is 125 m/55.0 s 5 2.27 m/s. You may have traveled at various speeds during the walk and, of course, you changed direction. Neither average velocity nor average speed provides information about these details. Q uick Quiz 2.1 Under which of the following conditions is the magnitude of the average velocity of a particle moving in one dimension smaller than the average speed over some time interval? (a) A particle moves in the 1x direction without reversing. (b) A particle moves in the 2x direction without reversing. (c) A par- ticle moves in the 1x direction and then reverses the direction of its motion. (d) There are no conditions for which this is true. Example 2.1   Calculating the Average Velocity and Speed Find the displacement, average velocity, and average speed of the car in Figure 2.1a between positions A and F.  2.2 Instantaneous Velocity and Speed 25 ▸ 2.1 c o n t i n u e d Solution Consult Figure 2.1 to form a mental image of the car and its motion. We model the car as a particle. From the position– time graph given in Figure 2.1b, notice that x A 5 30 m at t A 5 0 s and that x F 5 253 m at t F 5 50 s. Use Equation 2.1 to find the displacement of the car: Dx 5 x F 2 x A 5 253 m 2 30 m 5 283 m This result means that the car ends up 83 m in the negative direction (to the left, in this case) from where it started. This number has the correct units and is of the same order of magnitude as the supplied data. A quick look at Fig- ure 2.1a indicates that it is the correct answer. xF 2 xA Use Equation 2.2 to find the car’s average velocity: v x,avg 5 tF 2 tA 253 m 2 30 m 283 m 5 5 5 21.7 m/s 50 s 2 0 s 50 s We cannot unambiguously find the average speed of the car from the data in Table 2.1 because we do not have infor- mation about the positions of the car between the data points. If we adopt the assumption that the details of the car’s position are described by the curve in Figure 2.1b, the distance traveled is 22 m (from A to B) plus 105 m (from B to F), for a total of 127 m. 127 m Use Equation 2.3 to find the car’s average speed: v avg 5 5 2.5 m/s 50 s Notice that the average speed is positive, as it must be. Suppose the red-brown curve in Figure 2.1b were different so that between 0 s and 10 s it went from A up to 100 m and then came back down to B. The average speed of the car would change because the distance is different, but the average velocity would not change. 2.2 Instantaneous Velocity and Speed Often we need to know the velocity of a particle at a particular instant in time t rather than the average velocity over a finite time interval Dt. In other words, you would like to be able to specify your velocity just as precisely as you can specify your position by noting what is happening at a specific clock reading, that is, at some specific instant. What does it mean to talk about how quickly something is mov- ing if we “freeze time” and talk only about an individual instant? In the late 1600s, with the invention of calculus, scientists began to understand how to describe an object’s motion at any moment in time. To see how that is done, consider Figure 2.3a (page 26), which is a reproduction of the graph in Figure 2.1b. What is the particle’s velocity at t 5 0? We have already discussed the average velocity for the interval during which the car moved from position A to position B (given by the slope of the blue line) and for the interval during which it moved from A to F (represented by the slope of the longer blue line and calculated in Example 2.1). The car starts out by moving to the right, which we defined to be the positive direction. Therefore, being positive, the value of the average velocity during the interval from A to B is more representative of the ini- tial velocity than is the value of the average velocity during the interval from A to F, which we determined to be negative in Example 2.1. Now let us focus on the short blue line and slide point B to the left along the curve, toward point A, as in Figure 2.3b. The line between the points becomes steeper and steeper, and as the two points become extremely close together, the line becomes a tangent line to the curve, indicated by the green line in Figure 2.3b. The slope of this tangent line 26 Chapter 2 Motion in One Dimension x (m) 60 B 60 40 A C B B B 20 B 0 D 40 The blue line between 20 positions A and B E approaches the green 40 F tangent line as point B is A moved closer to point A. 60 t (s) 0 10 20 30 40 50 a b Figure 2.3 (a) Graph representing the motion of the car in Figure 2.1. (b) An enlargement of the Pitfall Prevention 2.2 upper-left-hand corner of the graph. Slopes of Graphs In any graph of physical data, the slope represents the ratio of the change in the represents the velocity of the car at point A. What we have done is determine the quantity represented on the verti- instantaneous velocity at that moment. In other words, the instantaneous velocity vx cal axis to the change in the quan- equals the limiting value of the ratio Dx/Dt as Dt approaches zero:1 tity represented on the horizontal axis. Remember that a slope has Dx units (unless both axes have the v x ; lim (2.4) S Dt 0 Dt same units). The units of slope in Figures 2.1b and 2.3 are meters In calculus notation, this limit is called the derivative of x with respect to t, written per second, the units of velocity. dx/dt: Dx dx v x ; lim 5 (2.5) Instantaneous velocity  Dt 0 Dt S dt The instantaneous velocity can be positive, negative, or zero. When the slope of the Pitfall Prevention 2.3 position–time graph is positive, such as at any time during the first 10 s in Figure 2.3, Instantaneous Speed and Instan- vx is positive and the car is moving toward larger values of x. After point B, vx is nega- taneous Velocity In Pitfall Pre- tive because the slope is negative and the car is moving toward smaller values of x. vention 2.1, we argued that the magnitude of the average velocity At point B, the slope and the instantaneous velocity are zero and the car is momen- is not the average speed. The mag- tarily at rest. nitude of the instantaneous veloc- From here on, we use the word velocity to designate instantaneous velocity. When ity, however, is the instantaneous we are interested in average velocity, we shall always use the adjective average. speed. In an infinitesimal time The instantaneous speed of a particle is defined as the magnitude of its instan- interval, the magnitude of the dis- placement is equal to the distance taneous velocity. As with average speed, instantaneous speed has no direction asso- traveled by the particle. ciated with it. For example, if one particle has an instantaneous velocity of 125 m/s along a given line and another particle has an instantaneous velocity of 225 m/s along the same line, both have a speed2 of 25 m/s. Q uick Quiz 2.2 Are members of the highway patrol more interested in (a) your average speed or (b) your instantaneous speed as you drive? Conceptual Example 2.2    The Velocity of Different Objects Consider the following one-dimensional motions: (A) a ball thrown directly upward rises to a highest point and falls back into the thrower’s hand; (B) a race car starts from rest and speeds up to 100 m/s; and (C) a spacecraft drifts through space at constant velocity. Are there any points in the motion of these objects at which the instantaneous velocity has the same value as the average velocity over the entire motion? If so, identify the point(s). 1Notice that the displacement Dx also approaches zero as Dt approaches zero, so the ratio looks like 0/0. While this ratio may appear to be difficult to evaluate, the ratio does have a specific value. As Dx and Dt become smaller and smaller, the ratio Dx/Dt approaches a value equal to the slope of the line tangent to the x-versus-t curve. 2 As with velocity, we drop the adjective for instantaneous speed. Speed means “instantaneous speed.”  2.2 Instantaneous Velocity and Speed 27 ▸ 2.2 c o n t i n u e d Solution (A) The average velocity for the thrown ball is zero because the ball returns to the starting point; therefore, its displace- ment is zero. There is one point at which the instantaneous velocity is zero: at the top of the motion. (B) The car’s average velocity cannot be evaluated unambiguously with the information given, but it must have some value between 0 and 100 m/s. Because the car will have every instantaneous velocity between 0 and 100 m/s at some time during the interval, there must be some instant at which the instantaneous velocity is equal to the average veloc- ity over the entire motion. (C) Because the spacecraft’s instantaneous velocity is constant, its instantaneous velocity at any time and its average velocity over any time interval are the same. Example 2.3    Average and Instantaneous Velocity A particle moves along the x axis. Its position varies with time according to x (m) the expression x 5 24t 1 2t 2, where x is in meters and t is in seconds.3 The 10 position–time graph for this motion is shown in Figure 2.4a. Because the 8 position of the particle is given by a mathematical function, the motion of Slope  4 m/s 6 the particle is completely known, unlike that of the car in Figure 2.1. Notice Slope  2 m/s D 4 that the particle moves in the negative x direction for the first second of motion, is momentarily at rest at the moment t 5 1 s, and moves in the posi- 2 tive x direction at times t. 1 s. A C 0 t (s) (A) Determine the displacement of the particle in the time intervals t 5 0 2