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3.2 Vector Operations. Review. Find the x and y component. Go 4 m to the right, 2 m up, 5 m down, 7m left, 3m to the left and 10 m up. X= 6m left y = 7m up. Pythagorean’s Theorem. c 2 = a 2 + b 2 c
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Review Find the x and y component. Go 4 m to the right, 2 m up, 5 m down, 7m left, 3m to the left and 10 m up. X= 6m left y = 7m up
Pythagorean’s Theorem c2 = a2 + b2 c a b
SOH CAH TOA • Sin q = opposite hypotenuse • Cos q = adjacent hypotenuse • Tan q = opposite adjacent
Helpful Hints to Solving Vector Problems • Place a dot on your paper • Draw the first vector given in the problem in the appropriate direction. • Without picking your pencil point off the paper draw the next vector in the appropriate direction. • Continue this with as many vectors as you have.
Helpful Hints to Solving Vector Problems 5. Draw your resultant from the dot in the beginning to the last vector arrow • Look at your drawing. You need to make a right triangle opposite your resultant. • Use pythagorean’s theorem to find the resultant side c 8. Vectors must have direction so you will use tan to find the angle from the resultant to the vector. (This angle always rotates around the original dot)
Direction N W E S
A plane travels from Huston, Texas to Washington, DC, which is 1540 km, East and 1160km north of Houston. What is the total displacement of the plane? C 1160 KM 1540 km
c2 = a2 + b2 c2 = (1540 km)2 + (1160km)2 C = 1928 km = 1930 km Tan q = 1160 q = 37.0o 1540 Answer: 1930 km, 37.0o north of east Practice: 90 degree Problems
Boat Problems A motorboat heads due east at 16 m /s across a river that flows due south at 9.0 m /s. • What Is the resultant velocity of the boat. • If the river is 136 m wide, how long does it take the motorboat to reach the other side? • How far downstream is the boat when it reaches the other side of the river?
What Is the resultant velocity of the boat? 16 m /s q c 9 m /s C 2 = (16)2 + (9)2 C = 18 m /s Tan q = 9 / 16 q = 30o 18 m /s, 30o south of east
If the river is 136 m wide, how long does it take the motorboat to reach the other side? V = d t t = d = 136 m V 16 m /s t = 8.50 s
How far downstream is the boat when it reaches the other side of the river? V = d t d = V t d = 9.0 m /s X 8.5 s = 77 m Practice: Boat problems
Components You are given the resultant (magnitude and direction) and you find the sides or components Use sin and cos to find the x and y component Hypotenuse angle y x
Component Example An arrow is shot from a bow at an angle of 25o above the horizontal with an initial speed of 45 m /s. Find the horizontal and vertical components of the arrow’s initial velocity. 45 m /s 25o y X
Components Sin 25o = y = 19 m /s 45 Cos 25o = x = 41 m /s 45 Practice problems on components
Putting it all together A mouse runs east for 4 m across the living room, turns 30o north of east and runs 2.8 m and finally heads due north 3 m to a hole in the wall. What is the resultant? 3 m 2.8 m 30o 4 m
Draw and examine. Look for a right triangle. The resultant is “from where you start to where you end it does not matter where you have been”. Find the components to make the right triangle. 3m C q 1.4 m 4m 2.4 m • Add up the all the x’s then all the y’s: x total = 6.4m y total = 4.4m • Use pythagorean’s theorem c 2 = a2 + b2 C = 7.8 m • tan q = 4.4 / 6.4 q = 35 o 7.8 m, 35 o North of East
A camper walks 4.5 km at 45o north of east then 4.5 km due south. Find the campers total displacement. 4.5 45o 4.5
4.5 sin 45 = x x = 3.184.5 sin 45 = y1= 3.184.5 - 3.18 = 1.32c2 = 3.182 + 1.322 c = 3.4tan q = 1.32 / 3.18 q= 22o3.4 m, 22o south of east 4.5 45o 4.5
Practice, Practice, Practice • Practice #1 • Practice #2 • Holt Text Pg 94 1-4 • QUIZ