New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product. reading assignment read chap 3

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1 New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product reading assignment read chap 3

2 Most physical quantities are described by a single number or variable examples: your age, your weight, today s temperature, the time, the color (frequency of light) of your car etc. The above are called scalars. Some physical quantities are better described by 2 or more numbers or variables. examples: displacement in 2 and 3 dimensions, you need magnitude and direction (two or more numbers) to completely describe it. These are called vectors, objects that require magnitude and direction Physical quantities that are vectors: include displacement, velocity, acceleration, force in 2 and 3 dimensions

3 Vectors, are more convenient and compact mathematical notation. Example. Suppose we want to describe the displacement from Glenwood Springs to Fort Collins. (note this is TWO dimensional displacement). We could say, the displacement is 100 miles East and 60 miles North. This is convenient for an automobile. OR we can represent this displacement by a vector drawn in red, denoted as A. To fully describe the vector we need to know is length (magnitude) and A its direction ( in this case the angle w.r.t. horizontal ). This is useful for airplanes Glenwood Springs Fort Collins Denver

4 displacement vector displacement vector from point A to B can be described by a magnitude (or length) and direction θ y axis magnitude or length B θ x axis A terminology: vector quantities are boldface characters, a r older texts use, a or a, regular font a or a is magnitude of vector a

5 UNIT VECTORS suppose we make unit vectors, 1 unit magnitude in the x direction, i, and 1 unit magnitude in the y direction, j. Y a x unit vectors, i y unit vectors j X a = i + i + i + i + i + j + j + j = 5 i + 3 j we can write vector a as a sum of vectors, in this case we add 5 i vectors and 3 j vectors. vector components of a are a x and a y. a x =5 and a y =3 and we can write a = a x i + a y j.

6 Self Test Question; Suppose we have vector A = 5 i and vector B = 5 j What is the magnitude of vector, C=A + B, or A + B?? Ans; What is the direction θ of the vector, A + B? Ans; B A

7 Self Test Question; Suppose we have vector A = 5 i and vector B = 5 j What is the magnitude of vector, C= A + B, or A + B?? Ans; A + B = sqrt ( ) = sqrt(50) What is the direction θ of the vector, A + B? Ans; θ = 45 C=A + B B A

8 Self Test Question; Suppose we have vector A = 5 i and vector B = 5 j What is the magnitude of vector, C= A + B, or A + B?? Ans; A + B = sqrt ( ) = sqrt(50) What is the direction θ of the vector, A + B? Ans; θ = 45 What are the x and y components of C? Ans; C=A + B A B

9 Self Test Question; Suppose we have vector A = 5 i and vector B = 5 j What is the magnitude of vector, C= A + B, or A + B?? Ans; A + B = sqrt ( ) = sqrt(50) What is the direction θ of the vector, A + B? Ans; θ = 45 What are the x and y components of C? Ans; C x = 5, C y = 5 C=A + B A B

10 We can write vectors in terms of magnitude and direction or in terms of the x component and the y component. What is the relation between these two different sets of variables? magnitude = a 2 2 x + ay = a = a tanθ = a y /a x or atan(a y /a x )=θ a x = a cosθ a y = a sinθ length a a y θ a x

11 adding vectors graphically, place the origin of one vector on the arrow tip of another a d e a = b + c + d + e c b

12 Multiplying vectors by real numbers (scalar), magnitude changes but direction does not. = i + i + i + i + i + i = 6 i = a Multiplying vector by a negative number, reverses direction, mag. same Subtracting vectors b = a = 6 i a + b a b a a - b -b

13 Scalar Product or dot product In Physics we will need to form a scalar quantity formed from two vectors. Later we will use them in Work and Electric fields definition: a b = a b cosθ, where θ is the angle between vectors. example, i i = 1 1 cos 0 = 1, i j = 1 1 cos90 =0 if a = a x i + a y j and b = b x i + b y j. a b = (a x i + a y j ) (b x i + b y j) = a b = a x b x i i + a x b y i j + a y b x j i + a y b y j j a b = a x b x + a y b y

14 Useful things to do with unit vectors and dot products. given any vector a, we can obtain the components by using the dot products with the unit vectors. a i =( a x i + a y j ) i = a x i i = a x We can say that the dot product projects the vector component in the direction of the unit vector. That is the dot product of a unit vector and a given vector will yield the vector component of the given vector. If we multiply a vector by itself then, a a = ( a x i + a y j ) ( a x i + a y j ) = a x2 + a y 2 = a 2 hence the length of a is sqrt. root of a a

15 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? Step 1 draw Diagram

16 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. N (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? v wind v plane θ v final W

17 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. N (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? v wind Solution step by step (1) final velocity eqn. is? v plane θ v final W

18 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. N (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? v wind Solution step by step (1) final velocity eqn. is? v final =v plane + v wind v plane θ v final (2) what is the angle: W

19 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. N (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? v wind Solution step by step (1) final velocity eqn. is? v final =v plane + v wind v plane θ v final W (2) what is the angle: sine of the angle θ between the velocity of the plane and north equals the ratios v wind and v plane sinθ = 90km/h / 200km/h = 0.45, θ = 26.7 (3) what is the plane velocity?

20 Problem: A plane is to fly due north. The speed of the plane relative to the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h. N (a) in which direction should the plane head? (b) how fast does the plane travel relative to the ground? v wind Solution step by step (1) final velocity eqn. is? v final =v plane + v wind v plane θ v final W (2) what is the angle: sine of the angle θ between the velocity of the plane and north equals the ratios v wind and v plane sinθ = 90km/h / 200km/h = 0.45, θ = 26.7 (3) what is the plane velocity? since v final and v wind are perpendicular, we use the Pythagorean Theorem to find the magnitude of v final, v plane2 = v wind2 + v final 2 v final = sqrt (v plane 2 - v wind 2 )

21 Self test A jet plane in straight horizontal flight passes over your head. When it is directly above you, the sound seems to come from a point behind the plane in a direction 30 from the vertical. The speed of the plane is: A) the same as the speed of sound B) half the speed of sound C) three-fifths the speed of sound D) times the speed of sound E) twice the speed of sound Hint Draw Picture and write down relevant features of the problem 2 minutes

22 Distance jet travels = v JET t Distance = v sound t Distance sound travels 30 Solution: 0.5 = sin(30 ) = v JET t / v sound t = v JET / v sound Ans) B