Grade 10 Physics Textbook PDF

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Ethiofetena.com Ethiopian No 1 Educational Website Physics Physics Physics Student Physics Physics Student Student Textbook FEDERAL FEDERALDEMOCRATIC FEDERAL DEMOCRATIC DEMOCRATIC REPUBLIC REPUBLIC REPUBLICOF OF ETHIOPIA OF ETHIOPIA ETHIOPIA MINISTRY MINISTRYOFOF MINISTRY EDUCATION OFEDUCATION EDUCATION Student Student StudentTextbook Textbook Textbook Textbook Textbook –– Grade Grade Grade10 Grade 10 10 – Grade Grade 10 10 10 FEDERAL FEDERALDEMOCRATIC FEDERAL DEMOCRATIC DEMOCRATIC REPUBLIC REPUBLIC OFOF REPUBLIC ETHIOPIA OFETHIOPIA ETHIOPIA MINISTRY MINISTRYOFOF EDUCATION EDUCATION MINISTRY OF EDUCATION Ethiofetena.com Ethiopian No 1 Educational Website Ethiofetena.com Ethiopian No 1 Educational Website Physics Student Textbook Grade 10 Writers Fikadu Eshetu (PhD) Mideksa Kasahun (MSc) Editors Moges Tsega (PhD), content editor Samuel Assefa (PhD), curriculum editor Felekech G/Egziabher (PhD), language editor Illustrator Umer Nuri (MSc) Designer Derese Tekestebrihan (PhD fellow) Evaluators Zafu Abraha (MSc) Girmaye Defar (MSc) Dessie Melese (MSc)  Ethiofetena.com Ethiopian No 1 Educational Website iv First Published in 2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education, under the General Education Quality Improvement Program for Equity (GEQIP-E) supported by the World Bank, UK’s Depart- ment for International Development/DFID-now merged with the Foreign, Common wealth and Development Office/FCDO, Finland Ministry for Foreign Affairs, the Royal Norwegian Embassy, United Nations Children’s Fund/UNICEF), the Global Partnership for Education (GPE), and Danish Ministry of Foreign Affairs, through a Multi Donor Trust Fund. © 2023 by the Federal Democratic Republic of Ethiopia, Ministry of Educa- tion. 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PHOTO CREDIT: ISBN: 978-99990-0-033-8  Ethiofetena.com Ethiopian No 1 Educational Website Contents 1 Vector Quantities 1 1.1 Scalars and Vectors....................... 2 1.2 Vector representations..................... 3 1.3 Vector addition and subtraction................ 6 1.4 Graphical method of vector addition............. 8 1.5 Vector resolution......................... 14 2 Uniformly Accelerated Motion 21 2.1 Position and Displacement................... 22 2.2 Average velocity and instantaneous velocity........ 25 2.3 Acceleration........................... 30 2.4 Equations of motion with constant acceleration...... 36 2.5 Graphical representation of uniformly accelerated motion 42 2.6 Relative velocity in one dimension.............. 50 3 Elasticity and Static Equilibrium of Rigid Body 59 3.1 Elasticity and plasticity..................... 60 3.2 Density and specific gravity.................. 63 3.3 Stress and Strain......................... 67 3.4 The Young Modulus....................... 72 3.5 Static equilibrium........................ 77 3.5.1 First condition of equilibrium............ 78 3.5.2 Second condition of equilibrium........... 79 4 Static and Current Electricity 91 4.1 Charges in Nature........................ 92 4.2 Methods of Charging a Body.................. 94 4.3 The electroscope......................... 97 4.4 Electrical Discharge....................... 99 4.5 Coulomb’s law of electrostatics................ 102 4.6 The electric field......................... 104 4.7 Electric circuits......................... 107 4.8 Current, Voltage, and Ohm’s Law............... 110 v  Ethiofetena.com Ethiopian No 1 Educational Website vi CONTENTS 4.9 Combination of resistors in a circuit............. 120 4.10 Voltmeter and ammeter connection in a circuit...... 128 4.11 Electrical safety in general and local context........ 132 4.12 Electric projects......................... 136 5 Magnetism 151 5.1 Magnet.............................. 152 5.2 Magnetic Field.......................... 155 5.3 The Earth’s magnetic field and the compass......... 159 5.4 Magnetic field of a current-carrying conductor....... 162 5.5 Magnetic force on a moving charge placed in a uniform magnetic field.......................... 165 5.6 Magnetic force on a current-carrying wire.......... 167 5.7 Magnetic force between two parallel current-carrying wires 170 5.8 Applications of magnetism................... 171 6 Electromagnetic Waves and Geometrical Optics 177 6.1 Electromagnetic (EM) waves.................. 178 6.2 EM Spectrum........................... 180 6.3 Light as a wave.......................... 186 6.4 Laws of reflection & refraction................. 190 6.5 Mirrors and lenses........................ 203 6.6 Human eye and optical instruments............. 228 6.7 Primary colors of light and human vision.......... 238 6.8 Color addition of light...................... 240 6.9 Color subtraction of light using filters............ 241  Ethiofetena.com Ethiopian No 1 Educational Website Unit 1 Vector Quantities Introduction In science, particularly in physics, you try to make measurements as pre- Brainstorming cise as possible. Several times in the history of science, precise mea- question: surements have led to new discoveries or important developments. Any List some phys- number or sets of numbers used for a quantitative description of a phys- ical quantities, ical phenomenon is called a physical quantity. Physical quantities can and classify them generally be divided in two groups: scalars and vectors. Scalars have only as scalars and magnitudes while vectors have both magnitude and direction. The con- vectors. cepts of vectors and scalars help us in understanding physics of different natural phenomena. You will learn about this topics in this unit. By the end of this unit, you should be able to: understand the differences between scalar and vector quantities; demonstrate vectors representation graphically; know how to add and subtract two or more vectors graphically; resolve a single vector into its components. 1  Ethiofetena.com Ethiopian No 1 Educational Website 2 Unit 1 Vector Quantities 1.1 Scalars and Vectors By the end of this section, you should be able to: define scalar and vector quantities; Exercise 1.1 describe the difference between vector and scalar quantities;  From what you learnt in grade 9, list some scalar and vector quantities. what do you think are the differences As discussed in the introduction, physical quantities can be classified into between vectors two categories. Physical quantities that fall in the first category are those and scalars? that can be completely specified by a number together with an appropri- ate unit of measurement. For instance, it makes perfectly good sense to say that the length of an object is 1.42 m or that the mass of an object is Key concepts 12.21 kg. You do not have to add anything to the description of length or mass. Similarly, the statement that the density of water is 1000 kg /m 3 is a  Scalars are complete description of density. Quantities that can be specified in this quantities that have only mag- simple and straightforward way are called scalar quantities. Thus, scalar nitude whereas quantities are physical quantities that can be completely specified by a Vectors are quan- single number together with an appropriate unit of measurement. Time, tities that have distance, speed, length, volume, temperature, energy and power are other both direction examples of scalar quantities. and magnitude. On the other hand, quantities that fall in the second category are those which require both magnitude and direction for their complete description. Exercise 1.2 A simple example is velocity. The statement that the velocity of a train is List examples of 100 km/h does not make much sense unless you also tell the direction in scalars and vec- which the train is moving. Force is another such quantity. You must specify tors other than not only the magnitude of the force but also the direction in which the those discussed in force is applied. Such quantities are called vectors. A vector quantity has the textbook. both magnitude and direction. Displacement, acceleration, momentum, impulse, weight and electric field strength are other examples of vector quantities.  Ethiofetena.com Ethiopian No 1 Educational Website 1.2 Vector representations 3 Section summary In physics, you deal generally with two kinds of quantities: scalars and vectors. Scalar are quantities that are specified only by their magni- tude while vectors are quantities that are specified by their magnitude and direction. Review questions 1. Explain how vector quantities differ from scalar quantities and give some examples for each. 2. Which of the following physical quantities are vectors and which are not: force, temperature, volume, velocity, age, weight? 1.2 Vector representations By the end of this section, you should be able to: Exercise 1.3 identify the magnitude and direction of a vector;  How can you represent vectors? discuss how vectors can be represented algebraically and graphi- cally; discuss about the different types of vectors. In the previous section, you learnt that vectors are represented both in mag- nitude and direction. On the other hand, vector quantities are represented either algebraically or geometrically. Algebraically, they are represented by a bold letter as A or with an arrow over the letter, for example,  A. For example, a displacement can be represented by the expression S = 50 km, Southwest. S = 50 km designates only the magnitude of the vector. The magnitude is also indicated by placing the absolute value notation around  Ethiofetena.com Ethiopian No 1 Educational Website 4 Unit 1 Vector Quantities the symbol that denotes the vector; so, you can write equivalently that S ≡ |S|. On the other hand, vectors are represented geometrically by an arrow, or an arrow-tipped line segment. Such an arrow having a specified length and direction shows the graphical representation of a vector. You will use this representation when drawing vector diagrams. The initial point of arrow is called tail and the final point of the arrow is the head as it is indicated in Figure 1.1. The arrow is drawn to scale so that its length represents the magnitude of the vector, and the arrow points in the specified direction of the vector. Hence, The length of the arrow represents the vector magnitude if it is drawn in scale. The arrow head represents the vector direction. Figure 1.1 Head and tail of a vector. Thus, in order to draw a vector accurately, you must specify a scale and include a reference direction in the diagram. To do this, you need a ruler to measure and draw the vectors to the correct length. The length of the arrow should be proportional to the magnitude of the quantity being repre- sented. So you must decide on a scale for your drawing. For example, you might let 1 cm on paper represent 2 N (1 cm represents 2 N), a force of 20 N towards the East would be represented as an arrow 10 cm long. A scale allows us to translate the length of the arrow into the vector’s magnitude. Key concepts The important thing is to choose a scale that produces a diagram of rea- sonable size. A reference direction may be a line representing a horizontal  Vectors are surface or the points of a compass. represented either algebraically or geometrically. The following are the procedures that you might use for drawing vectors graphically. 1. Decide upon a scale and write it down. 2. Determine the length of the arrow representing the vector by using the scale.  Ethiofetena.com Ethiopian No 1 Educational Website 1.2 Vector representations 5 3. Draw the vector as an arrow. Make sure that you fill in the arrow Activity 1.1 head. By using a ruler, 4. Fill in the magnitude of the vector. a protractor, and a square paper, Example 1.1 graphically draw Draw the vector, 16 km East, to scale by indicating the scale that you have the following nu- used: merical vectors. Solution A) 10 cm , 60o First, let us decide upon a scale. Let 1 cm represent 4 km. So if 1 cm = 4 km, B) 15 cm , 120o then 16 km = 4 cm and the direction is in the East. Using this information, C) 3 cm , 30o you can draw the vectors as arrows as follows. Figure 1.2 A scaled diagram for a vector with a magnitude of 16 km. On the other hand, the following are some of the different types of vectors. 1. Zero vector or Null vector: a vector with zero magnitude and no direction. 2. Unit Vector: vector that has magnitude equal to one. 3. Equal vectors: vectors that have the same magnitude and same direction. Figure 1.3 Two equal vectors. 4. Negative of a vector: a vector that have the same magnitude but opposite direction with the given vector. Section summary Algebraically, a vector is represented by a bold face letter or an arrow over the letter. Geometrically, a vector is represented by an arrow where the length of the arrow represents magnitude and the arrow head represents the direction for the vector.  Ethiofetena.com Ethiopian No 1 Educational Website 6 Unit 1 Vector Quantities Review questions 1. Give an example of a vector stating its magnitude, units and direction. 2. Choose your own scale and draw arrows to represent the fol- lowing vectors: A)  A = 40 km North,  = 32 m/s making an angle of 60o with the horizontal. B) B 3. Discuss the different types of vectors. 1.3 Vector addition and subtraction By the end of this section, you should be able to: Exercise 1.4 explain how to add and subtract vectors; Is it possible to add two vectors in define the term resultant vector. the same way as you did in scalars? Different mathematical operations can be performed with vectors. You Explain. need to understand the mathematical properties of vectors, like addition and subtraction. Addition of Vectors The addition of scalar quantities is non problematic, it is a simple arith- Key concepts metic sum. For example, the total mass of 2 kg plus 3 kg is 5 kg. The increase in temperature from 5o C to 12o C is 7o C. However, like scalars,  Scalars and you cannot add two vectors. This is because when two vectors are added, vectors can never you need to take account of their direction as well as their magnitude. Of be added.  For any two course, you should remember that only vectors of the same kind can be vectors to be added. For example, two forces or two velocities can be added. But a force added, they must and a velocity cannot be added. be of the same nature. The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together. In other words, the  Ethiofetena.com Ethiopian No 1 Educational Website 1.3 Vector addition and subtraction 7 individual vectors can be replaced by the resultant where the overall effect is the same. If vectors   have a resultant R, A and B  this can be represented mathemati- cally as, = R  A +B (1.1) Figure 1.4 Addition of vectors  A  and B. Subtraction of Vectors Vector subtraction is a straight forward extension of vector addition. If you want to subtract B from A, written   , you must first define what A −B is meant by subtraction. As it is discussed in the previous section, the  is defined to be −B negative of vector B  ; that is, graphically the negative of any vector has the same magnitude but opposite in direction as shown in  has the same length as −B Figure 1.5. In other words, B  , but points in the opposite direction. Essentially, you just flip the vector so that it points in the opposite direction.  from vector  The subtraction of vector B A is then simply defined to be the   addition of −B to A. That is,  and the Figure 1.5 Vector B . negative of Vector B  = A −B ) A + (−B (1.2) Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results. Hence, as it is indicated in Figure  from the tip of  1.6, draw vector −B A and join the tail of A with the tip of  , then the resulting vector is the difference (  −B  ). A-B Section Summary Vector addition is a means of finding the resultant of a number  Figure 1.6 Subtraction of B of vectors.  from A. Subtraction of a vector is addition of the negative of a vector.  Ethiofetena.com Ethiopian No 1 Educational Website 8 Unit 1 Vector Quantities Review questions Exercise 1.5 1. What is meant by subtraction of vector? If two vectors have equal mag- 2. What is meant by resultant vector? nitude, what are the maximum and minimum 1.4 Graphical method of vector addition magnitudes of their sum? By the end of this section, you should be able to: describe the graphical method of vector addition; use the graphical method of vector addition to solve problems. You can add or subtract vectors using the algebraic or graphical method of vector addition. You will learn about the algebraic method in your grade 11 physics. In this section, you will learn about the graphical method of vector addition and subtraction. As discussed above, one method for adding vectors involves manipulating their graphical representations on paper. So using the graphical method of vector addition, vectors are drawn to scale and the resultant is determined using a ruler and protractor. The following is the discussion of the primary graphical techniques: the triangle method, the parallelogram method, and the polygon method. Procedure for using graphical method of vector addition Key concepts Graphically, vec- Decide on an appropriate scale. Record it on the diagram. tors can be added Pick a starting point. using the triangle, parallelogram and Draw first vector with appropriate length and in the indicated polygon method direction. of vector addition. Draw the second and remaining vectors with appropriate  Ethiofetena.com Ethiopian No 1 Educational Website 1.4 Graphical method of vector addition 9 length and direction. Draw the resultant based on the specific rule you are using. Measure the length of the resultant; use the scale to convert to the magnitude of the resultant. Use a protractor to measure the vector’s direction. Triangle method of vector addition Triangle law of vector addition is used to find the sum of two vectors. This law is used to add two vectors when the first vector’s head is joined to the tail of the second vector and then joining the tail of the first vector to the head of the second vector to form a triangle, and hence obtain the resultant sum vector. That’s why the triangle law of vector addition is also called the head-to-tail method for the addition of vectors. Thus, if two vectors acting simultaneously on a body are represented both in magnitude and direction by two sides of a triangle taken in an order, then the resultant vector (both magnitude and direction) of these two vectors is given by the third side of that triangle taken in the opposite order. This is the statement for the triangle law of vector addition. Consider two vectors   shown in Figure 1.7 (a). To add these two A and B vectors using the triangle method, the head of vector  A should be joined . Then, the resultant vector R to the tail of vector B  has its tail at the tail of   as shown in Figure 1.7 (b). This is written Figure 1.7 The triangle rule for A and its head at the head of B the addition of two vectors. as: = R  A +B (1.3)  Ethiofetena.com Ethiopian No 1 Educational Website 10 Unit 1 Vector Quantities Activity 1.2 Consider adding two vectors   graphically. The two vectors A and B are shown in Figure 1.8. Using the above procedure of vector addi- tion, add these two vectors using the triangle law of vector addition if the angle θ is 30o. Figure 1.8 Two vectors  . A and B Parallelogram method of vector addition The vector addition may also be understood by the law of parallelogram. The parallelogram law of vector addition is used to add two vectors when the vectors that are to be added form the two adjacent sides of a parallel- ogram by joining the tails of the two vectors. Then, the sum of the two vectors is given by the diagonal of the parallelogram. Thus, if two vectors are represented by the two adjacent sides (both in magnitude and direction) of a parallelogram drawn from a point, then their resultant vector is represented completely by the diagonal of the parallelogram drawn from the same point. This is the statement for the parallelogram law of vector addition Suppose two vectors   are at arbitrary positions as shown in Figure A and B 1.9 (a). Translate either one of them in parallel to the beginning of the other Figure 1.9 The Parallelogram vector, so that after the translation, both vectors have their origins at the rule for the addition of two same point. Now, at the end of vector   A you draw a line parallel to vector B vectors.  you draw a line parallel to vector  and at the end of vector B A (the dashed lines in Figure 1.9 (b)). In this way, you obtain a parallelogram. From Activity 1.3 the origin of the two vectors, you draw a diagonal of the parallelogram as  of the two vectors. shown in (Figure 1.9 (b)). The diagonal is the resultant R Repeat activity 1.2 using the parallel- Thus, ogram method of vector addition. = R  A +B (1.4)  Ethiofetena.com Ethiopian No 1 Educational Website 1.4 Graphical method of vector addition 11 Since vector addition is commutative,  =B A +B + A (1.5) Polygon method of vector addition This law is used for the addition of more than two vectors. According to this law, if you have a large number of vectors, place the tail end of each successive vector at the head end of previous one. The resultant of all vectors can be obtained by drawing a vector from the tail end of first to the head end of the last vector. Suppose you want to draw the resultant vector R  of four vectors  , A, B  , and D C  shown in Figure 1.10 (a). You select any one of the vectors as the first vector and make a parallel translation of a second vector to a position where the origin ("tail") of the second vector coincides with the Figure 1.10 The polygon rule end ("head") of the first vector. Then, you select a third vector and make for the addition of vectors. a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. You repeat Activity 1.4 this procedure until all the vectors are in a head-to-tail arrangement like Use the polygon the one shown in (Figure 1.10 (b)). You draw the resultant vector R  by method of vec- connecting the origin ("tail") of the first vector with the end ("head") of the tor addition to last vector. The end of the resultant vector is at the end of the last vector. find the resultant  of the vector R Thus, the resultant vector R  is an arrow drawn from the tail of vector D  three vectors:  , i.e., R to the head of vector B =D  + +B A +C  as shown in Figure 1.10 A = 25.0m, 49.00  (b). Because the addition of vectors is associative and commutative, you North of East, obtain the same resultant vector regardless of which vector you choose to  = 23.0m, 15.00 B be first, second, third, or fourth in this construction. North of East and  = 32.0m, 68.00 C Note: You will follow the same procedure during the subtraction of vectors. South of East. Let us now consider a few special cases of addition of vectors. Choose a reason- able scale. 1. When the two vectors are in the same direction (parallel to each other)  Ethiofetena.com Ethiopian No 1 Educational Website 12 Unit 1 Vector Quantities If vectors  A and B are parallel, then the magnitude of the resultant  is the sum of the magnitudes of the two vectors. Hence, the vector R magnitude of the resultant vector is |R| = |A + B | (1.6) Since the two vectors are in the same direction, the direction of the resultant vector is in the direction of one of the two vectors. Key concepts To determine the resultant of two vectors acting: in the same direction, add the given vectors and take the common direction. in opposite directions, get the difference and take the direc- tion of the vector with the greater value.  Figure 1.11 Resultant vector R 2. When the two vectors are acting in opposite directions   of two vectors A and B when If vectors   are anti-parallel (i.e., in opposite direction), then A and B they are (a) same direction and  is the difference of the mag- the magnitude of the resultant vector R (b) opposite direction. nitudes of the two vectors. Hence, the magnitude of the resultant vector is |R| = |A − B | (1.7) Since the two vectors are in opposite directions with one another, the direction of the resultant vector is in the direction of the larger vector. Note: The resultant of two vectors acting on the same point is maxi- mum when the vectors are acting in the same direction and mini- mum when they act in opposite directions. 3. When the two vectors are perpendicular If vectors   are perpendicular to each other as shown in 1.12, A and B  is obtained using the then the magnitude of the resultant vector R  Ethiofetena.com Ethiopian No 1 Educational Website 1.4 Graphical method of vector addition 13 Pythagoras theorem. Hence, the magnitude of the resultant vector is  |R| = A2 + B 2 (1.8) The direction of the resultant vector is obtained using the trigono- metric equation:   −1 B θ = t an (1.9) A Note: You can compare the result you obtain in each of the three cases with a ruler and protractor. Surely, you will obtain similar Figure 1.12 Two perpendic- result. ular vectors  A and B ; and its  resultant vector R. Example 1.1 Exercise 1.6 Two vectors have magnitudes of 6 units and 3 units. What is the magnitude If two vectors  A of the resultant vector when the two vectors are (a) in the same direction,  and B are per- (b) in opposite direction and (c) perpendicular to each other? pendicular, how you can find the Solution: sum of the two You are given with two vectors of magnitudes 6 units and 3 units. vectors? (a) When the two vectors are in the same direction, |R| = (6 + 3) units = 9 units. (b) When the two vectors are in the opposite directions, |R| = (6 − 3) units = 3 units. (c) When the two vectors are perpendicular to each other,   |R| = A 2 + B 2 = ( 62 + 32 ) units = 6.7 units. Please compare the result you obtained here with the value you obtained with the direct measurement by a ruler. Section Summary Two vectors can be added by graphical means using the tri- angle and the parallelogram method. But for more than two vectors, the polygon method is used.  Ethiofetena.com Ethiopian No 1 Educational Website 14 Unit 1 Vector Quantities Review questions 1. Two vectors   have the same magnitude of 5 units and A and B  points to the North East and  they start from the origin: B A . What points to the South West exactly opposite to vector B would be the magnitude of the resultant vector? Why? 2. If two vectors have equal magnitude, what are the maximum and minimum magnitudes of their sum? 3. If three vectors have unequal magnitudes, can their sum be zero? Explain. 4. Consider six vectors that are added tail-to-head, ending up where they started from. What is the magnitude of the resul- tant vector? 5. Vector C is 6 m in the x-direction. Vector D  is 8 m in the  + D. y-direction. Use the parallelogram method to work out C  1.5 Vector resolution By the end of this section, you should be able to: Exercise 1.7 What do you resolve a vector into horizontal and vertical components; think is vector find the resultant of two or more vectors using the component resolution? method. In the previous discussion of vector addition, you saw that a number of vectors acting together can be combined to give a single vector (the resultant). In much the same way, a single vector can be broken down into a number of vectors when added give the original vector. These vectors which sum to the original are called components of the original vector. The process of breaking a vector into its components is called resolving into components.  Ethiofetena.com Ethiopian No 1 Educational Website 1.5 Vector resolution 15 Placing vectors in a coordinate system that you have chosen makes it possible to decompose them into components along each of the chosen coordinate axes. In the rectangular coordinate system shown in Figure 1.13, vector  A is broken up or resolved into two component vectors. One, A x , is parallel to the x-axis, and the other, A y , is parallel to the y-axis. The horizontal and vertical components can be found by two methods: graphical method and simple trigonometry. Let us look at them one by one. Graphical method of vector resolution: The following are the steps that you follow to resolve a vector graphically. 1. Select a scale and draw the vector to scale in the appropriate direc- tion. 2. Extend x- and y-axes from the tail of the vector to the entire length of the vector and beyond. 3. From the arrow head of the vector, construct perpendicular projec- tions to the x- and the y-axes. 4. Draw the x-component from the tail of the vector to the intersection of the perpendicular projection with the x-axis. Label this compo- nent as A x. Figure 1.13 The horizontal (A x ) and vertical (A y ) components 5. Draw the y-component from the tail of the vector to the intersection of vector  A. of the perpendicular projection with the y-axis. Label this compo- nent as A y. 6. Measure the length of the two components and use the scale to determine the magnitude of the components. Trigonometric method of vector resolution: The trigonometric method of vector resolution relies on an understanding of the sine, cosine, and tangent functions. You can find the components by using trigonometry. The components are calculated according to these  Ethiofetena.com Ethiopian No 1 Educational Website 16 Unit 1 Vector Quantities equations, where the angle θ is measured counterclockwise from the posi- tive x-axis. Adjacent side Ax cos θ = = =⇒ A x = Acos θ (1.10) hypotenuse A and Opposite side Ay sin θ = = =⇒ A y = Asi n θ (1.11) hypotenuse A You can see that the original vector is the sum of the two component Key concepts vectors. Vectors can be  x + Ay A=A (1.12) resolved into com- Because A x and A y are at a right angle (90o ), the magnitude of the resultant ponents along a vector can be calculated using the Pythagorean Theorem. specified set of coordinate’s axes.  |A| = Ax 2 + A y 2 (1.13) To find the angle or direction of the resultant, recall that the tangent of the angle that the vector makes with the x-axis is given by the following.   −1 Ay Opposite side side Ay θ = t an where tan θ = = (1.14) Ax Adjacent side Ax Example 1.2 A motorist undergoes a displacement of 250 km in a direction 30o North of East. Resolve this displacement into its components. Solution: Draw a rough sketch of the original vector. You can use trigonometry to calculate the magnitudes of the components (along North and along East) of the original displacement: Figure 1.14 Resolving of resul- tant displacement. S N = (250)(si n 30o ) = 125 km S E = (250)(cos 30o ) = 216.5 km  Ethiofetena.com Ethiopian No 1 Educational Website 1.5 Vector resolution 17 Please check your answer with the graphical method, i.e., with ruler and protractor. Example 1.3 A boy walks 3 km due East and then 2 km due North. What is the magnitude and direction of his displacement vector? Solution: You first make an overhead view of the boy’s movement as shown in Figure Figure 1.15 The magnitude and direction of the boy’s motion. 1.15. The magnitude of the displacement |S| is given by the Pythagorean theorem as follows: |S| = [(3 km)2 + (2 km)2 ]1/2 = 3.6 km The direction that this displacement vector makes relative to east is given by: 2 km t an θ = = 0.666 3 km Thus, θ = t an −1 (0.666)=33.69o Hence, the boy’s displacement vector is 3.61 km with 56.31o East of North, or 33.69o North of East. Exercise 1.8 Could a vector Please check your answer with the graphical method, i.e., with ruler and ever be shorter protractor. than one of its components? Section summary Could it be equal Any vector can be resolved into a horizontal and a vertical in length to one of component. its components? Explain. The combined effect of the horizontal and vertical compo- nents of the vector quantity is the same as the original vector.  Ethiofetena.com Ethiopian No 1 Educational Website 18 Unit 1 Vector Quantities Review questions 1. What is resolution of a vector? 2. Draw simple vector diagrams and resolve them into their com- ponents. (a) 40 N at an angle of 30o from the horizontal. (b) 10 m/s at an angle of 80o from the horizontal. (c) 1900 km at an angle of 40o from the vertical. 3. A car travels 10 km due North and then 5 km due West. Find graphically and analytically the magnitude and direction of the car’s resultant vector. 4. A girl walks 25.0o North of East for 3.10 km. How far would she have to walk due North and due East to arrive at the same location? Virtual Lab On the soft copy of the book, click on the following link to per- form virtual experiment on vector quantities unit under the guid- ance of your teacher. 1. Vector Addition PhET Experiment. End of unit summary Scalar is a quantity specified only by its magnitude. Vectors is a quantity specified by its magnitude and direction. The vector is represented by an arrow drawn at a suitable scale where:  Ethiofetena.com Ethiopian No 1 Educational Website 1.5 Vector resolution 19 – The arrow length represents the vector magnitude. – The arrow head represents the vector direction. Vector addition is nothing but finding the resultant of a num- ber of vectors acting on a body while vector subtraction is addition of the negative of a vector. The sum of two or more vectors is called the resultant. The resultant of two vectors can be found using either the paral- lelogram method or the triangle method. For more than two vectors, one can use the polygon method of vector addition. The method of finding the components of vectors is called resolving vector. When vector A is decomposed along the rectangular coordi- nate system, the horizontal component of  x = Acos θ A is A and the vertical component of  A is Ay = Asi n θ. End of unit questions and problems 1. A vector drawn 15 mm long represents a velocity of 30 m/s. How long should you draw a vector to represent a velocity of 20 m/s? 2. A vector that is 1 cm long represents a displacement of 5 km. How many kilometers are represented by a 3 cm vector drawn to the same scale? 3. Describe how you would add two vectors graphically. 4. Which of the following actions is permissible when you are graphically adding one vector to another? A) move the vector B) rotate the vector C) change the vector’s length. 5. In your own words, write a clear definition of the resultant of two or more vectors.  Ethiofetena.com Ethiopian No 1 Educational Website 20 Unit 1 Vector Quantities 6. Explain the method you would use to subtract two vectors graphically. 7. You walk 30 m South and 30 m East. Find the magnitude and direction of the resultant displacement both graphically and algebraically. 8. A hiker walks 14.7 km at an angle 35o East of South. Find the East and North components of this walk. 9. If two vectors have equal magnitudes, can their sum be zero? Explain. 10. Based on the three vectors in Figure 1.16, which of the follow- ing is true? (a)   +C A +B  =0 (b)   +B A =C  + (c) C  A=B Figure 1.16 Magnitude and 11. For the two vectors   with magnitude 6.8 cm and 5.5 A and B direction of three vectors cm in Figure 1.17, determine the magnitude and direction of: = (a) R  A +B = (b) R  A −B =B (c) R − A Figure 1.17 Magnitude and 12. Three vectors   and C A, B  have a magnitude and direction of direction of two vectors 21 unit North, 16 unit East and 26 unit South, respectively. Graphically determine the resultant of these three vectors? 13. If vx = 9.8 m/s and vy = 6.4 m/s, determine the magnitude and direction of  v.  Ethiofetena.com Ethiopian No 1 Educational Website Unit 2 Uniformly Accelerated Motion Introduction In your grade 9 Physics, you have learnt that uniform motion occurs when Brain storm- an object moves at a steady speed in a straight line. Most moving objects, ing question however, do not display uniform motion. Any change in an object’s speed In your everyday or direction or both causes the motion of an object to become non uni- life, you come form. This non uniform motion, or changing velocity, is called accelerated across a range of motion. A car ride in a city at rush hour during which the car must speed constant accel- up, slow down, and turn corners is an obvious example of accelerated eration motions. motion. In this unit, you will learn about uniformly accelerated motion. Can you give two examples for such By the end of this unit, you should be able to: type of motion? know terms that are used to describe uniformly accelerated motion; understand the different types of motions used to describe physical phenomena; know the equation of motions that describe the motion of an object under uniform acceleration; solve motion problems using uniformly accelerated formulas; understand relative velocity in one dimension. 21  Ethiofetena.com Ethiopian No 1 Educational Website 22 Unit 2 Uniformly Accelerated Motion 2.1 Position and Displacement By the end of this section, you should be able to: define terms such as position, displacement, and distance; determine the distance and displacement traveled by an object; describe the difference between distance and displacement. Position Exercise 2.1 To describe the motion of an object, you must first be able to describe its Can two objects position, or where it is at any particular time. The word position describes be at the same your location (where you are). However, saying that you are here is mean- distance from a ingless, and you have to specify your position relative to a known reference single point while point. So, you need to specify its position relative to a convenient frame of being in different reference. A frame of reference is an arbitrary set of axes from which the positions? Why or position and motion of an object are described. why not? To visualize position for objects moving in a straight line, you can imagine the object is on a number line. The object may be placed at any point on the number line in the positive numbers or the negative numbers. It is common to choose the original position of the object to be on the zero Key Concept mark as shown in Figure 2.1. In making the zero mark the reference point,  The location you have chosen a frame of reference. The exact position of an object is of an object in a the separation between the object and the reference point. frame of reference is called position. Position is thus the location of an object with reference to an origin. It can be negative or positive. It has units of length: centimeter (cm), meter (m) or kilometer (km). For example, depending on that reference point you choose, you can say that the new constructed school is 300 m from Kemal’s house (with Kemal’s house as the reference point or origin).  Ethiofetena.com Ethiopian No 1 Educational Website 2.1 Position and Displacement 23 Displacement From your grade 9 Physics, you know that distance is the total length of the path taken in going from the initial position to the final position. Distance is a scalar. But the difference between the initial and final position vectors of a body is called its displacement. Basically, displacement is the shortest distance between the two positions and has a certain direction. Thus, displacement is a vector quantity. In Figure 2.1, distance is the length of the dashed line, while displacement is the straight-line distance from the starting point to the endpoint. Activity 2.1 Figure 2.1 The possible dis- On a piece of graph paper, draw a scale map of your home and tance and displacement of an school area. Determine your displacement and estimate the dis- object in motion between two points. tance you travel: a) from home to school. b) from school to home. If the initial position (s o ) from which an object moves to a second position (s) in a particular frame of reference, then the displacement ∆s  can be written as: Exercise 2.2  = s − so ∆s (2.1) What is the dif- In order to answer exercise 2.2, consider the motion of an object moving ference between along a straight path. The object starts its journey from O which is treated distance and displacement? as its reference point as shown in Figure 2.2. Let A, B, and C represent the position of the object at different instants. Figure 2.2 Positions of an object on a straight line path.  Ethiofetena.com Ethiopian No 1 Educational Website 24 Unit 2 Uniformly Accelerated Motion Exercise 2.3 For motion of the object from O to A, the distance covered is 60 km and the magnitude of displacement is also 60 km. During its motion from O to Can the mag- nitude of the A and back to B, the distance covered = 60 km + 25 km = 85 km while the displacement of magnitude of displacement = 60 km - 25 km = 35 km. Thus, the magnitude an object from its of displacement (35 km) is not equal to the path length (85 km). Further, original position you will notice that the magnitude of the displacement for a course of ever exceed the motion may be zero, but the corresponding distance covered is not zero. total distance If you consider the object to travel back to O, the final position coincides moved? Explain. with the initial position, and therefore, the displacement is zero. However, the distance covered in this journey is OA + AO = 60 km + 60 km = 120 km. Thus, the two different physical quantities (distance and displacement), are used to describe the overall motion of an object and to locate its final position with reference to its initial position at a given time. Example 2.1 A person walks 70 m East, and then 30 m West. Find the displacement. Solution: The displacement of a person walking 70 m to the East and then turning Figure 2.3 A person walking 70 around and walking back (West) a distance of 30 m as shown in Figure 2.3 m East and then 30 m West.  = s − s o = 70 m − 30 m as both vectors are in an opposite direction is: ∆s with one another. Thus, Key Concepts ∆→ −s = 40 m E ast  Distance is the actual path The displacement is only 40 m since the person is now only 40 m from the that is travelled starting point but the total distance traveled is 70 m + 30 m = 100 m. by a moving body, where as displacement is the change in position (final position minus initial position).  Ethiofetena.com Ethiopian No 1 Educational Website 2.2 Average velocity and instantaneous velocity 25 Section summary A description of motion depends on the reference frame from which it is described. Position is the location of an object compared to a reference Activity 2.2 frame (point). Walk from one The distance an object moves is the length of the path along corner of your which it moves. classroom to its opposite corner Displacement is the difference between the initial and final along its sides. positions of an object. Measure the dis- tance covered Review questions by you and mag- nitude of the 1. Explain the difference between position and displacement. displacement. 2. Give an example that clearly shows the difference among dis- What difference tance traveled, displacement, and magnitude of displacement. would you notice between the two Identify each quantity in your example. in this case? 3. A body travels a distance of 15 m from A to B and then moves a distance of 20 m at right angles to AB. Calculate the total distance traveled and the displacement. 2.2 Average velocity and instantaneous velocity Exercise 2.4 Can the average By the end of this section, you should be able to: speed ever equal define instantaneous and average velocity of a body in motion; the magnitude of the average describe the difference between average velocity and instantaneous velocity? If "no," velocity; why not? If "yes," solve problems related to the average velocity. give an example.  Ethiofetena.com Ethiopian No 1 Educational Website 26 Unit 2 Uniformly Accelerated Motion Average velocity In grade 9, you learnt that the rate of change of distance with time is called speed, while the rate of change of displacement is known as velocity. Un- like speed, velocity is a vector quantity. Key Concept When an object travels a certain distance with different velocities, its motion is specified by its average velocity. The average velocity of a body Velocity is the is defined as the body’s displacement (∆s) divided by the time interval physical quantity (∆t ) during which that displacement occurs. Let s o and s be its positions that describes at instants t o and t, respectively. You can express average velocity (vav ) how a moving object’s displace- mathematically as: ment changes.  ∆s s − s o vav = = (2.2) ∆t t − t o Exercise 2.5 where t − t o is change in time, and t o is the starting time which is com-  Does the monly zero. speedometer of a car measure speed or velocity? The SI unit for average velocity is meters per second (m/s or m s −1 ). But there are also many other units, such as kilometer per hour (km/h), miles per hour (mi/h (also written as mph)) and centimeter per second (cm/s) in Exercise 2.6 common use.  Describe how the instantaneos The average speed of an object is obtained by dividing the total distance velocity differs traveled by the total time taken: from the averag total distance travelled stot velocity? vav = = (2.3) total time taken ttot  In which sit- uation will the If the motion is in the same direction along a straight line, the average instantaneous speed is the same as the magnitude of the average velocity. However, this velocity and av- is always not the case. erage velocity of an object be the same?  Ethiofetena.com Ethiopian No 1 Educational Website 2.2 Average velocity and instantaneous velocity 27 Instantaneous velocity Suppose the magnitude of your car’s average velocity for a long trip was 20 m/s. This value, being an average, does not convey any information about Key Concept how fast you were moving or the direction of the motion at any instant Instantaneous during the trip. Both can change from one instant to another. Surely, there velocity is the ve- were times when your car traveled faster than 20 m/s and times when it locity at a specific traveled more slowly. instant in time (or over an infinitesi- The instantaneous velocity of the car indicates how fast the car moves and mally small time the direction of the motion at each instant of time. Thus, it is the rate of interval). change in displacement as change in time approaches zero. Mathemati- cally, the instantaneous velocity ( v ) of a body is given by s − so v=  when t − to approaches 0 (2.4) t − to The magnitude of the instantaneous velocity of a moving car is the reading of the speedometer. Road traffic accidents are among the main causes of mortality in Ethiopia. Speed is still the most common factor in fatal road accidents, accounting Figure 2.4 A fatal car accident. for more than half of all road deaths each year. Figure 2.4 shows one of the fatal car crash where at least 5 people were died and 17 others injured Exercise 2.7 after a minibus collided with a parked car somewhere in Ethiopia. It may  Can you imag- not seem like much, but driving even a few kilometers per hour above ine the things that the speed limit greatly increases the risk of an accident. Speed limits are would happen if used to set the legal maximum or minimum speed at which road vehicles a driver does not may travel on a given stretch of road. They are generally indicated on a obey the speed traffic sign reflecting the maximum or minimum speed permitted that is limits set and expressed usually in kilometers per hour (km/h). Speed limits are being are moving in started to be monitored by traffic officers in the various streets and roads uniformly acceler- of our country. Speed limits are used to regulate the speed of vehicles in ated motion? certain places and it also controls the flow of traffic. It is also observed to minimize accidents from happening.  Ethiofetena.com Ethiopian No 1 Educational Website 28 Unit 2 Uniformly Accelerated Motion Example 2.2 It takes you 10 minutes to walk with an average velocity of 1.2 m/s to the North from the bus stop to the museum entrance. What is your displace- ment? Solution: m You are given with ∆t = 10 minutes = 600 s and v av = 1.2 s , North. You want to find ∆s. Since  ∆s m v av =  ,  = vav × ∆t = 1.2 ∆s × 600 s = 720 m, Nor t h ∆t s This means the displacement has a magnitude of 720 m and a direction to the North. Example 2.3 A passenger in a bus took 8 s to move 4 m to a seat on provided place forward. What is his average velocity? Solution:  = 4 m and ∆t = 8 s. You are given ∆s What you want to find is Vav. The average velocity is thus  ∆s 4m v av =  = = 0.5 m/s ∆t 8s Example 2.4 A car travels at a constant speed of 50 km/h for 100 km. It then speeds up to 100 km/h and is driven another 100 km. What is the car’s average speed for the 200 km trip?  Ethiofetena.com Ethiopian No 1 Educational Website 2.2 Average velocity and instantaneous velocity 29 Solution: You are given s 1 = 100 km, v 1 = 50 km/h, s 2 = 100 km, v 2 = 100 km/h. In order to find the average speed, you first need to find the total distance traveled and total time taken. Thus, the total distance traveled is 100 km + 100 km = 200 km. The total time taken is t 1 + t 2 where s1 100 km ∆t 1 = = = 2h v 1 50 km/h s2 100 km ∆t 2 = = =1h v 2 100 km/h The car’s average speed is thus Exercise 2.8 total distance travelled 200 km Cheetahs, the vav = = = 66.7 km/h. total time taken 3h world’s fastest 50 km/h + 100 km/h land animals, can Note: Averaging the two speeds ( ) gives you a wrong 2 run up to about answer which is 75 km/h. The average speed of an object is obtained by 125 km/h. A chee- dividing the total distance traveled by the total time taken. tah chasing an impala runs 32 m Section summary north, then sud- denly turns and Average velocity is change in displacement divided by time runs 46 m west taken. before lunging at the impala. The Instantaneous velocity is the velocity of an accelerating body entire motion at a specific instant in time. takes only 2.7 s. (a) Determine the The magnitude of instantaneous velocity is its instantaneous cheetah’s average speed. speed for this Review questions motion. (b) Determine the 1. How do you find the average velocity of an object in motion cheetah’s average between two points? velocity.  Ethiofetena.com Ethiopian No 1 Educational Website 30 Unit 2 Uniformly Accelerated Motion 2. Explain the difference between average speed and average velocity? 3. There is a distinction between average speed and the magni- tude of average velocity. Give an example that illustrates the difference between these two quantities. 4. If an object has the instantaneous velocity of 20 m/s to East, what is its instantaneous speed? 5. A car moves with an average velocity of 48.0 km/h to the East. How long will it take him to drive 144 km on a straight high- way? 6. An athlete runs 12 km to the North, then turns and runs 16 km to the East in three hours. a) What is his/her displacement? b) Calculate his/her average velocity. c) Calculate average speed. Exercise 2.9 1) If a body has constant velocity 2.3 Acceleration on straight level surface, what is By the end of this section, you should be able to: the magnitude of explain acceleration in one dimension; its acceleration? distinguish between instantaneous acceleration and average accel- 2) Does the di- eration; rection of ac- celeration be in calculate the average acceleration. the direction of velocity itself? While traveling in a bus or a car, you might have noticed that sometimes its speed increases and sometimes it slows down. That is, its velocity changes with time. The quantity that describes the rate of change of velocity in a given time interval is called acceleration. Any change in velocity whether positive, negative, directional, or any combination of these is acceleration.  Ethiofetena.com Ethiopian No 1 Educational Website 2.3 Acceleration 31 In everyday conversation, to accelerate means to speed up. Thus, the greater the acceleration is, the greater the change in velocity over a given time is. Average acceleration When you watch the first few seconds of a liftoff, a rocket barely seems to move. With each passing second, however, you can see it move faster until it reaches an enormous speed. How could you describe the change in the rocket’s motion? When an object changes its motion, it is accelerating. The magnitude of the average acceleration is defined by the change in an object’s velocity divided by the time interval in which the change occurs. That is, Change in velocity Average acceleration = time taken v v − vo ∆ a=  = (2.5) ∆t t − to where v o is the initial velocity of an object and v is the final velocity of an object at instants t o and t, respectively. In this equation, t − t o is the length of time over which the motion changes. In SI units, acceleration has units of meters per second squared (m/s 2 ). The direction of average acceleration is the direction of change in velocity. Key Concept If an object speeds up, the acceleration is in the direction that the object is  Acceleration moving. You get on a bicycle and begin to pedal. The bike moves slowly at occurs whenever first, and then accelerates because its speed increases. When the speed an object speeds of an object increases, it is said to be accelerating. On the other hand, if up, slows down, or an object slows down, the acceleration is opposite to the direction that changes direction. the object is moving. This is commonly referred to as deceleration. In Figure 2.5, a light train in Addis Ababa, Ethiopia, decelerates as it comes into a station. Thus, the train is accelerating in a direction opposite to its direction of motion.  Ethiofetena.com Ethiopian No 1 Educational Website 32 Unit 2 Uniformly Accelerated Motion Figure 2.5 A decelerating light train. When you speed up, your final speed will always be greater than your initial speed. So subtracting your initial speed from your final speed gives Exercise 2.10 a positive number. As a result, your acceleration is positive when you  What do you are speeding up. When you slow down, the final speed is less than the mean by an in- initial speed. Because your final speed is less than your initial speed, your stantaneous ac- acceleration is negative when you slow down. celeration for an object in motion? Instantaneous acceleration The object moving in a straight line may undergo an increase, or decrease Key Concept in acceleration or it may move with a uniform acceleration or zero acceler- Instantaneous ation. Thus, in such cases, the average acceleration does not describe the acceleration is motion of the object at every instant. The average acceleration only pro- the average ac- vides the mean value of the acceleration instead of the actual acceleration celeration at a of the object during the motion while the instantaneous acceleration gives specific instant the exact acceleration at every instant during the motion. in time (or over an infinitesimally Instantaneous acceleration is a quantity that tells us the rate at which small time inter- an object is changing its velocity at a specific instant in time anywhere val).  Ethiofetena.com Ethiopian No 1 Educational Website 2.3 Acceleration 33 along its path. Hence, instantaneous acceleration a, or acceleration at a specific instant in time, is obtained using the same process discussed for instantaneous velocity. That is, you calculate the average velocity between two points in time separated by ∆t and let ∆t approach zero. You see that average acceleration a = ∆v ∆t approaches instantaneous acceleration as ∆t approaches zero. The instantaneous acceleration is, thus mathematically expressed as, Exercise 2.11 ∆v  For what type a= as ∆t approaches 0. (2.6) ∆t of motion does the average and For uniformly accelerated motion, the instantaneous acceleration has the instantaneous same value as the average acceleration. acceleration be the same? Example 2.5 A car accelerates on a straight road from rest to 75 km/h in 5 s. What is the magnitude of its average acceleration? Solution: You are given with v 0 = 0, v = 75 km/h and ∆t = 5 s. You want to find the average acceleration. Exercise 2.12  Think about The average acceleration can be calculated by the greatest ac- v − v o 75 km/h − 0 km/h km/h celerations you a av =  = = 15. t − to 5 s − 0s s have experienced. Where did they This is read as "fifteen kilometers per hour per second" and means that, occur? Did they on average, the velocity changed by 15 km/h during each second. That involve speeding is, assuming the acceleration was constant, during the first second, the up or slowing car’s velocity increased from zero to 15 km/h. During the next second, its down? What ef- velocity increased by another 15 km/h, reaching a velocity of 30 km/h at fects did they t = 2.0 s, and so on. This result contains two different time units: hours have on you? and seconds. You usually prefer to use only seconds. To do so, you can change km/h to m/s: 75 km 75 km 1000 m 1h =( )( )( ) = 21 m/s h h 1 km 3600 s  Ethiofetena.com Ethiopian No 1 Educational Website 34 Unit 2 Uniformly Accelerated Motion v v − vo ∆ a av =  = ∆t t − to 21 m/s − 0m/s m/s a av =  = 4.2 = 4.2 m/s 2 5 s − 0s s Note that acceleration tells us how quickly the velocity changes, whereas velocity tells us how quickly the position changes. Exercise 2.13  Discuss the Example 2.6 concept of de- An automobile is moving to the