CHAPTER 1INTRODUCTION AND MATHEMATICAL CONCEPTSCONCEPTUAL QUESTIONS____________________________________________________________________________________________1. REASONING AND SOLUTION a. The SI unit for x is m. The SI units for the quantity vt are ms(s) mms(s) mFHGIKJTherefore, the units on the left hand side of the equation are consistent with the units on theright hand side.b. As described in part a, the SI units for the quantities x and vt are both m. The SI units forthe quantity 12at2 arems(s ) m22FHGIKJTherefore, the units on the left hand side of the equation are consistent with the units on theright hand side.c. The SI unit for v is m/s. The SI unit for the quantity at isms(s)ms2FHGIKJTherefore, the units on the left hand side of the equation are consistent with the units on theright hand side.d. As described in part c, the SI units of the quantities v and at are both m/s. The SI unit ofthe quantity 12at3 isms(s ) m s23FHGIKJ 2 INTRODUCTION AND MATHEMATICAL CONCEPTSThus, the units on the left hand side are not consistent with the units on the right hand side.In fact, the right hand side is not a valid operation because it is not possible to add physicalquantities that have different units. e. The SI unit for the quantity v3 is m3/s3. The SI unit for the quantity 2ax3 isms(m )ms2232FHGIKJTherefore, the units on the left hand side of the equation are not consistent with the units onthe right hand side.f. The SI unit for the quantity t is s. The SI unit for the quantity 2 xa ism(m / s )msms s222FHGIKJ Therefore, the units on the left hand side of the equation are consistent with the units on theright hand side.____________________________________________________________________________________________2. REASONING AND SOLUTION The quantity tan  is dimensionless and has no units.The units of the ratio x/v arem(m / s)msmsFHGIKJThus, the units on the left side of the equation are not consistent with those on the right side,and the equation tan  = x/v is not a possible relationship between the variables x, v, and . ____________________________________________________________________________________________3. REASONING AND SOLUTION It is not always possible to add two numbers that havethe same dimensions. In order to add any two physical quantities they must be expressed inthe same units. Consider the two lengths: 1.00 m and 1.00 cm. Both quantities are lengthsand, therefore, have the dimension [L]. Since the units are different, however, these twonumbers cannot be added. ____________________________________________________________________________________________4. REASONING AND SOLUTIONChapter 1 Conceptual Questions 3 a. The dimension of a physical quantity describes the physical nature of the quantity and thekind of unit that is used to express the quantity. It is possible for two quantities to have thesame dimensions but different units. All lengths, for example, have the dimension [L].However, a length may be expressed in any length unit, such as kilometers, meters,centimeters, millimeters, inches, feet or yards. As another illustration, the quantities 100 g and 1.5 kg are masses and have thedimensions [M]; however, they have different units.b. All quantities with the same units must have the same dimensions. For example, allquantities expressed in kilograms have the dimension [M]; all quantities expressed in metershave the dimensions [L]. ____________________________________________________________________________________________5. REASONING AND SOLUTION For the equation to be valid, the dimensions of the lefthand side of the equation must be the same as the dimensions on the right hand side. Sincethe quantity c has no dimensions, it does not contribute to the dimensions of the right handside, regardless of the value of n. Therefore, the value of n cannot be determined fromdimensional analysis.____________________________________________________________________________________________6. REASONING AND SOLUTION The following table shows the value of sin , cos , the ratio (sin )/(cos ) and tan .sin  cos  (sin )/(cos ) tan 30.00.500 0.866 0.577 0.57739.0 0.629 0.777 0.810 0.81053.0 0.799 0.602 1.33 1.3360.0 0.866 0.500 1.73 1.73From the definitions given in Equations 1.1-1.3, we havesin//tan cos   h hh hhhoaoa ____________________________________________________________________________________________7. REASONING AND SOLUTION a. The graph below shows sin  plotted on the vertical axis and  on the horizontal axis, with  in 15° increments from  = 0 to  = 720°.4 INTRODUCTION AND MATHEMATICAL CONCEPTS.72066060054048042036030024018012060-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0Sin(degre es)b. The graph below shows cos  plotted on the vertical axis and  on the horizontal axis, with  in 15° increments from  = 0 to  = 720°..72066060054048042036030024018012060-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0(degre es)Cos____________________________________________________________________________________________8. REASONING AND SOLUTION A vector quantity has both magnitude and direction.The number of people attending a football game, the number of days in a month, and thenumber of pages in a book can all be completely specified by giving a magnitude only.Hence, none of these quantities can be considered a vector.____________________________________________________________________________________________Chapter 1 Conceptual Questions 5 9. REASONING AND SOLUTION For two vectors to be equal, they must be equal inmagnitude and have the same direction. Only vectors A, B, and D have the same magnitude.These three vectors are shown below:NorthEast30°ANorthEast30°BNorthEast60°DClearly, only vectors A and D have the same magnitude and point in the same direction;hence, A and D are equal.____________________________________________________________________________________________10. REASONING AND SOLUTION For two vectors to be equal, they must be equal inmagnitude and have the same direction. Thus, two vectors with the same magnitude are notnecessarily equal. They must also point in the same