Vectors Pearland ISD Physics Warm-up: Have students do “Introduction to Orienteering” as soon as class starts.

Acknowledgements © 2013 Mark Lesmeister/Pearland ISD This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.

Acknowledgements Select questions taken from Serway and Faughn, Holt Physics © 2002 by Holt, Rinehart and Winston.

Vectors and scalars Vectors are quantities that have both a direction and a magnitude. Examples include: Displacement Velocity Acceleration Force Quantities that have only a magnitude are called scalars. “The directions you followed in the activity were examples of vectors.” (1)

Representing vectors Vectors can be represented by words. “Take your team 2 ‘clicks’ (km) north” “US Air 45, new course 30o at 500 mph.” Vectors can be represented by symbols. In the text, boldface indicates vectors. Examples: So far, we have represented vectors with words. (1) (2)

Representing vectors Vectors can be represented graphically. The direction of the arrow is the direction of the vector. The length of the arrow tells the magnitude Vectors can be moved parallel to themselves and still be the same vector. Vectors only tell amount and direction, so a vector doesn’t care where it starts. We can also represent vectors graphically. Follow directions under “Resolving vectors” in the lab. (1) Since vectors are just magnitude and direction, not where they start, we can move them around and they are still the same vector. (2) magnitude and direction, can we “add” them just by adding the numbers? What would happen if we just added all the numbers in our list of vectors in the lab?

Multiplying and dividing by scalars Multiplying or dividing a vector by a scalar results in a new vector. Multiplying or dividing by a positive number changes the magnitude of the vector but not the direction. Multiplying or dividing by a negative number changes the magnitude and reverses the direction. What if I said, I was driving North at 30 mph. When I passed the cop, I tripled my speed. So the magnitude of the velocity changed, but not direction. V became 3 V (4)

Vector operations

Adding vectors The sum of two vectors is called the resultant. To add vectors graphically, draw each vector to scale. Place the tail of the second vector at the tip of the first vector. Vectors can be added in any order. To subtract a vector, add its opposite. (1) With vectors, it doesn’t mean anything to just add up the magnitudes. In the lab, you couldn’t just add up all the steps you took. It mattered which direction you took. In the lab, you found the final *result* when you applied each vector in turn. That one final result could have taken the place of our individual vectors. This is what it means to add vectors. We want the one vector that has the same *result* as our individual vectors taken together. In the lab, you should have seen how to add vectors. First, what did you do? You moved one vector, and then, from the endpoint of that one, you moved to the next vector, and so on. How could we draw this? (Draw it out on board.) Notice that I am doing the displacements tip to tail. I start vector 2 where vector 1 ended, etc. (2) (3) In your lab, did it matter what order you added vectors? Different route, but same ultimate ending point. (4) Now, subtraction is how we “undo” addition. If we add 3, we can subtract 3 to get back to original number. How do we undo adding a vector? (5)

Vector Addition Practice A vector of magnitude 30 units is added to a vector of magnitude 50 units. Which of the following could possible be the magnitude of the resultant? A) 10 units B) 15 units C) 27 units D) 88 units

Graphical addition of vectors A hiker follows the following directions. She goes 7 kilometers at 300 , then goes 5 kilometers in a direction of 3150 . (That’s the same as 450 west of north.) A thunderstorm threatens, and she needs to get back to the starting point. What path should she follow? Add these displacements graphically to find out. Let’s add these two vectors up graphically first.

Graphical Vector Addition Practice Using a protractor, ruler and a piece of paper (or mini-whiteboard), add the following vectors by making a scale drawing. 30 m/s E + 20 m/s at 30o N of East. 20 m S + 20 m at 60o W of South.

Adding perpendicular vectors Perpendicular vectors can be easily added. Use the Pythagorean theorem to find the magnitude of the resultant. Use the tangent function to find the direction of the resultant. What if we want to add vectors without having to draw them. Point out to students in the lab, question 3 asks how to find the magnitude of resultant vector if DELx = +6 and DEL Y = +8. These vectors are perpendicular, i.e. lie along x and y. (1) Using Pythagorean theorem, can find length of hypotenuse. Draw for students. (2) How do we get angle? Remember, tan angle = length of opposite side/length of adjacent Find magnitude and direction of angle from lab. Then do Practice 3A, #2 as GP.

Adding Perpendicular Vectors Practice From Holt Physics, p. 91 “While following the directions on a treasure map, a pirate walks 45.0 m North, then turns and walks 7.5 m East. What single straight line displacement could the pirate have taken?” “Emily passes a soccer ball 6.0 m directly across the field to Kara, who then kicks the ball 14.5 m directly down the field to Luisa. What is the ball’s displacement as it travels between Emily and Luisa?”

Resolving vectors into components. Any vector can be resolved, that is, broken up, into two vectors, one that lies on the x-axis and one on the y-axis. Explore: Point out that with displacement vectors, following x and y is same as direct route. This also applies to other kinds of vectors. A 3 m/s north wind and a 4 m/s east wind have same effect as 5 m/s NE wind. Now, if we know the magnitude and direction of the overall vector, can we find “components”, that is, x and y vectors that together will add up to the overall vector? Modeling: (Do above practice assuming angle is 20 degrees, and length is 6 m/s.)

Resolving Vectors An arrow is shot from a bow at an angle of 25 degrees above the horizontal, with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow’s initial velocity. GP

Resolving Vectors v=45m/s vy q=25o vx GP: Above IP: Practice 3B, #1,3,5,7

Adding non-perpendicular vectors Resolve each vector into x and y components, using sin and cos. Add the x components together to get the total x component. Add the y component together to get the total y component. Find the magnitude of the resultant using Pythagorean theorem. Find the direction of the resultant using the inverse tan function. First, sketch situation. Then, resolve vectors into components. Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately. Next we find the magnitude of the resultant

Adding non-perpendicular vectors Resolving Vector 1: Δx1 Δy1 30o D1 = 7 km First, sketch situation.

Adding non-perpendicular vectors Resolving Vector 1: X (km) Y D1 3.5 6 D2 DR First, sketch situation. Then, resolve vectors into components.

Adding non-perpendicular vectors Resolving Vector 2: D2 = 5 km Δy2 Δx2

Adding non-perpendicular vectors Resolving Vector 2: X (km) Y D1 3.5 6 D2 -3.5 DR Then, resolve vectors into components.

Adding non-perpendicular vectors Add the x components to get the resultant x component. Use the Pythagorean Theorem to find the magnitude of the resultant. Use the inverse tan function to find the direction of the resultant. X (km) Y D1 3.5 6 D2 -3.5 DR 9.5