North by Northwest - An Introduction to Vectors

Transcription

1 HPP A9 North by Northwest - An Introduction to Vectors Exploration GE 1. Let's suppose you and a friend are standing in the parking lot near the Science Building. Your friend says, "I am going to run at a speed of 5 [m/s]. Guess where I will be in 10 [s]?" 1. Would you be able to answer your friend's question? Are you missing an important piece of information? Invention Quantities that have direction, as well as magnitude, are called vectors and can be represented as arrows where the length of the arrow represents the magnitude and the arrow gives the direction. Quantities that do not have direction are called scalars. GE Which of the following are vectors and which are scalars? a) Wind speed 15mph b) Wind 15mph NE c) Calories in a donut d) Push e) Temperature f) Gradient of a hill g) Where the science building is, relative to where you live. h) How far away the science building is from where you live. 2. Would velocity for one-dimensional motion be considered a scalar or vector? 3. How about acceleration? Supported in part by NSF-CCLI Program under grants DUE # and DUE #

2 HPP A9 2 Invention: How do you add vectors? For one dimension Graphically: tail-to-head + = = 3 units 5 units 3 units + 5 units 8 units The sum of vectors is called the resultant vector (resultant for short). resultant has a magnitude of 8 units and it is pointing to the right. In the case above the As on number line: Vector going one direction (say to the right) is taken to be on number line pointing in positive direction. Vector going in opposite direction is considered to be pointing in negative direction. Adding vectors going in same direction is like adding positive numbers. Adding vectors going in opposite directions is like adding positive and negative numbers i.e. subtraction + = = 3 units + (- 5 units) = 3 units 5 units = (- 2 units) Like adding or subtracting regular numbers the order does not matter. GE Draw the resultant vector of the following. Give its magnitude and direction.

3 HPP A = 8 units 4 units 5 units Invention - Adding vectors in two dimensions Vectors in the plane can be added in a similar fashion to the linear case shown above. Suppose we want to add vectors A and B shown below. A B We can add them graphically be taking B and moving it over to A, placing the tail of B at the tip of A. The vector sum is drawn from the tail of A to the tip of B. A B A + B

4 HPP A9 4 GE Add the displacement vectors A and B graphically. A B 2. What is the magnitude of the vector sum? 3. What is the direction of the vector sum, measured from the horizontal? Exploration The figure below shows a displacement vector for a person who goes from the origin to the coordinates (4 [m], 5 [m]). Y X GE Is walking directly from (0,0) to (4,5) the only path a person could take?

5 HPP A If the person takes an indirect path from (0,0) to (4,5), is the displacement vector the same? Suppose there is a large pile of boxes in the direct path as shown below. Y X 3. What would be a simple way to bypass the boxes and arrive at the same destination? Invention Any vector in the plane can be visualized as the addition of a vector in the x direction and a vector in the y direction. A = A x + A y The figure below shows this vector addition graphically.

6 HPP A9 6 Y A = A x + A y A y A x X The vector in the x direction, A x, is called the x-component. The vector in the y direction, A y, is called the y-component. The magnitude of the components can be obtained from the original vector by using trigonometry. Y θ X A x = Acos(θ ) and A y = Asin(θ ) GE Find the x and y components of vector A shown below. It has a length of 5.0 [m/s].

7 HPP A9 7 55º A x = A y = Invention If you know the components you can use trigonometry to find the magnitude of the vector and the angle it makes. 2 2 A 1 y A = A x + A y θ = tan Ax It is always a good idea to draw a picture pointing out which angle you calculate, to avoid any ambiguity. Invention Vector addition can be done very precisely using components. If you know the components of two vectors A and B, then the vector sum C is determined by getting its components as shown below. C x = A x + B x C y = A y + B y

8 HPP A9 8 A B GE Find the x and y components of vector A and B in the above figure. You can use a ruler to measure lengths. A x = A y = B x = B y = 2. Calculate the vector sum C of A and B using components. C x = C y = 3. Calculate the magnitude of C using the components. 4. Calculate the angle that C makes with the positive x-direction, using components. 5. Compare your magnitude and direction of C with what you obtained graphically in GE 4.