Not everything moves in a perfectly straight line. Some things zig and then zag.
Not everything moves in a perfectly straight line. Some things zig and then zag. In this episode we take a look at the vector addition and you learn a way to organize the vector information you are provided with.(7:45) We also take a look at how to report a resultant vector’s magnitude and direction. (3:35) Finally, we delve into resolving vectors into their component vectors using trigonometry. (6:00)
The Question of the Day asks: (9:28)
How should the arrows representing vectors be drawn when adding to find a resultant?
a) tip to tail b) tip to tip c) tail to tail
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Hi and welcome to the APsolute Recap: Physics 1 Edition. Today’s episode will focus on vector addition and resolving vectors into their components.
Lets Zoom out:
Unit 1 – Kinematics
Topic 1.1
Big Idea – Force Interactions
When airline pilots fly their planes, they are very concerned with their “heading”. You may take a flight that heads due north for 100 kilometers and later takes a heading of 30 degrees counterclockwise from the east for 200 km. Being able to add these vectors helps pilots know where they are located.
Lets Zoom in:
First and foremost, you will want to have your calculator in degree mode since we will be using angle measurements. When it comes to working with vectors, we are going to either be adding vectors together and forming resultants, or we will be taking a resultant and breaking it into x and y direction components.
Let’s start with vector addition. Assuming that vectors are along the same axis, we are literally going to sum positive and negative vectors to find that direction’s component of the resultant. To show you what I mean, let’s assume being a nice son or daughter, you volunteer to do the grocery shopping for your household this week. A very… likely scenario… maybe. After entering the store, you walk north for 15 meters down the cereal aisle, but quickly realize you passed your favorite cereal and you walk south 7 meters to find it. Your north/south displacement was 8 meters north or 8 meters in the positive y-direction. That would be the y-component of your displacement.
Moving down the list of essential food items, you head east for 12 meters to find the milk. Then you head west for 24 meters. Your x-displacement is 12 meters west, or -12 meters. You now have the x-component of your displacement from the location you entered the grocery store.
The problem is, you cannot just sum x and y direction vectors. Although we call it vector addition, we need to use the Pythagorean Theorem to add components that are along different axes. So, we should find that the magnitude of your displacement from the moment you entered the store is 14.42 meters. Well, I know it sounds like we have done a lot of work to figure this out already, but it still isn’t enough information to tell a friend where to find you if they are looking for you in the store.
“Hey, yeah, so when I walked through the doors, I walked 14.42 meters.” See? It isn’t descriptive enough, so we also need to tell our friend which direction to walk. We need to know an angle. In order to figure out just which angle makes sense, you should sketch the components on graph paper. I like to start with a point labeled “i” for initial position, and then I draw an arrow representing the x-component, in this case an arrow pointing west that is 12 units long. Picking up where you left off, use the tip of the first vector to start the tail of the y-component which will be an arrow pointing northward and will be 8 units long. Since that is where your friend should look for you, you should label that point with an “f” indicating that it was your final position. It should be noted that when adding vectors, you should always draw the tip of one arrow touching the tail of the other.
Now, you can connect the point labeled “i” to the point labeled “f” with a dotted line that has an arrow pointed in the direction your friend would need to walk to find you. In this case, that direction would be west-ish and north-ish. This dotted line arrow is known as the resultant. You should see that we have now formed a triangle with a hypotenuse that is 14.42 units long. If you do the inverse tangent function of the ratio 8/12, you will find that the angle nearest to the initial position is 33.7 degrees and you could now tell your friend to head 14.42 meters at an angle of 33.7 degrees north of west, or clockwise from west. Some people like to standardize how they report resultant vectors, and the most common way, is the angle measurement when rotating counterclockwise from the +x axis (east). For your grocery store motion, that would mean an angle of 146.3 degrees. Counterclockwise is considered the positive direction for rotation.
The beauty of learning about vector addition is that it doesn’t just work for displacements like our grocery store scenario. It works with ALL vectors! Displacements, velocities, accelerations, forces, etc.
The other skill you should master is the ability to take a resultant vector and to break it into its x and y components vectors. For example, while shooting hoops with a friend, you break out a radar gun or speed gun and measure the velocity of your friend’s shot to be 13 m/s. Your friend shoots the ball at an angle of 30 degrees above horizontal. But, what was the initial x-velocity? What was the initial y-velocity? We can answer these questions by doing a bit of the opposite of what we did earlier. We can draw an arrow representing the 13 m/s shot at an angle of 30 degrees. From the tail of that arrow, we can draw the x-component and then connect to the tip of that x-component arrowhead, make a turn and draw an arrow pointed upward to represent the y-component. This second component should close the shape and we should now have a triangle that we can solve for the x and y legs using trigonometric functions. For the x direction, we can use cosine 30 degrees and set it equal to the ratio of adjacent (x-velocity) over hypotenuse (13 m/s). You will find that the x-velocity is 11.3 m/s. In the y-direction you can use the sine function to find the y-velocity to be 6.5 m/s. This process is known as resolving a vector into its components and it is SUPER helpful.
Now! Let’s put it all together! Going back to the example of the pilot, we head north 100 km and then 200 km at an angle of 30 degrees counterclockwise from east. This is getting complicated! Have no fear! You can organize your work with a table that consists of two columns, one for the x components and one for the y components. The table should have 3 rows for each of the 2 vectors (V1 and V2) and then a final row for the sums of the components. For vector 1 we have an x component of 0 and a y component of +100 km since it moved north only. For vector 2 we would have to resolve that vector into its x and y components using trig. You will find that the x component is +173.2 km and the y-component is + 100 km. You can now sum all x components to arrive at +173.2 km and sum all y components to arrive at +200 km. Using the Pythagorean Theorem, you can find the magnitude of the resultant to be 264.6 km. After sketching the x and y components of the resultant and connecting with a dotted line you can see the right triangle that has the resultant pointed at an angle of 49.1 degrees counterclockwise from east.
To recap……
Vectors along the same axis can literally be added. When on different axes, vectors are added with trigonometry. We use arrows to represent the magnitude and direction of vectors. We can add x and y vector components together to form a resultant, and we can break resultants into components by using trigonometric functions. Regardless of complexity, there is an organized way to keep track of vectors and to sum them.
Coming up next on the APsolute RecAP Physics 1 Edition, we will be utilizing what we have learned about kinematics and vectors to investigate projectile motion.
Today’s Question of the Day focuses on vector addition.
Question: How should the arrows representing vectors be drawn when adding to find a resultant?
a) tip to tail b) tip to tip c) tail to tail