Physics is the Science of Measurement We begin with the measurement of length: its magnitude and its direction. Length Weight Time

Distance: A Scalar Quantity A scalar quantity: Contains magnitude only and consists of a number and a unit. A B  Distance is the length of the actual path taken by an object. distance = 20 m

Displacement—A Vector Quantity A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 30 0 ) A B D = 12 m, 20 o Displacement is the straight-line separation of two points in a specified direction.Displacement is the straight-line separation of two points in a specified direction. 

Distance and Displacement Net displacement: 4 m,E 6 m,W D What is the distance traveled? 10 m !! D = 2 m, W Displacement is the change of position based on the starting point. Consider a car that travels 4 m, E then 6 m, W.Displacement is the change of position based on the starting point. Consider a car that travels 4 m, E then 6 m, W. x= +4 x = +4 x= -2 x = -2

Identifying Direction A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.) 40 m, 50 o N of E EW S N 40 m, 60 o N of W 40 m, 60 o W of S 40 m, 60 o S of E Length = 40 m 50 o 60 o

Identifying Direction Write the angles shown below by using references to east, south, west, north. EW S N 45 o EW N 50 o S Click to see the Answers S of E 45 0 W of N

Rectangular Coordinates Right, up = (+,+) Left, down = (-,-) (x,y) = (?, ?) x y (+3, +2) (-2, +3) (+4, -3) (-1, -3) Reference is made to x and y axes, with + and - numbers to indicate position in space

Trigonometry Review Application of Trigonometry to VectorsApplication of Trigonometry to Vectors y x R  y = R sin  x = R cos  R 2 = x 2 + y 2 Trigonometry

Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30 o. 90 m 30 0 The height h is opposite 30 0 and the known adjacent side is 90 m. h h = (90 m) tan 30 o h = 57.7 m

Finding Components of Vectors A component is the effect of a vector along other directions. The x and y components of the vector (R,  are illustrated below. x y R  x = R cos  y = R sin 

Example 2: A person walks 400 m in a direction of 30 o N of E. How far is the displacement east and how far north? x y R  x = ? y = ? 400 m   E N The y-component (N) is OPP: The x-component (E) is ADJ: x = R cos  y = R sin  E N

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? x = R cos  x = (400 m) cos 30 o = +346 m, E x = ? y = ? 400 m   E N Note: x is the side adjacent to angle 30 0 ADJ = HYP x Cos 30 0 The x-component is: R x = +346 m

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? y = R sin  y = (400 m) sin 30 o = m, N x = ? y = ? 400 m   E N OPP = HYP x Sin 30 0 The y-component is: R y = +200 m Note: y is the side opposite to angle 30 0

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? R x = +346 m R y = +200 m 400 m   E N The x- and y- components are each + in the first quadrant Solution: The person is displaced 346 m east and 200 m north of the original position.

Resultant of Perpendicular Vectors Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord. R is always positive;  is from + x axis x y R 

Example 3: A woman walks 30 m, W; then 40 m, N. Find her total displacement.  = 59.1 o N of W (R,  ) = (50 m, o ) R = 50 m -30 m +40 m R 

Component Method 1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others. 2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant. 3. Write each vector in x,y components. 4. Add vectors algebraically to get resultant in x,y components. Then convert to the total vector (R,  ).

Example 4. A boat moves 2.0 km east then 4.0 km north, then 3.0 km west, and finally 2.0 km south. Find resultant displacement. EN 1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others. 2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant. Note: The scale is approximate, but it is still clear that the resultant is in the fourth quadrant. 2 km, E A 4 km, N B 3 km, W C 2 km, S D

Example 4 (Cont.) Find resultant displacement. 3. Write each vector in i,j notation: A = +2 x B = + 4 y C = -3 x D = - 2 y 4. Add vectors A,B,C,D algebraically to get resultant in x,y components. R =R =R =R = -1 x + 2 y. 1 km, west and 2 km north of origin. EN 2 km, E A 4 km, N B 3 km, W C 2 km, S D 5. Convert to resultant vector See next page.

Example 4 (Cont.) Find resultant displacement. EN 2 km, E A 4 km, N B 3 km, W C 2 km, S D Resultant Sum is: R = -1 x + 2 y R y = +2 km R x = -1 km R  Now, We Find R,  R = 2.24 km  = N of W

Conclusion of Chapter 3B - Vectors