The Physics Philes, lesson 3: A Journey to Vector-ian Times
In which a tutor is procured, lines are drawn, and more time is allotted.
Remember last week when there was so much math? So. Much. Math. Well, for this week at least, we can forget that noise. We’re gonna draw some vectors! (And do a little math. But trust me. It’s not too bad.)
These are vectors:
Vectors, it is becoming clear, are just super important in physics, so it’s important to know what they are and how they function. A vector is a quantity that has direction and a magnitude. A magnitude is the size of the vector. For example, 10 m/s east is a vector; it has a direction (east) and a magnitude (30 m/s.) Don’t get confused. If you were to say that the car was traveling at 30 m/s, that isn’t a vector because there is no direction. (That is an example of a scalar quantity.)
Pretty simple so far, right? But you know what would be really cool? If we could add vectors together. I will make it so.
Adding vectors isn’t like adding scalar quantities like 3 or 7. When we add vectors what we are trying to come up with is a particle’s (or car’s or person’s) displacement, or distance from the starting location to the ending location.
Let’s add vector A and vector B. (For the purposes of this post, I’m just going to type out the whole word, vector. I know that I’m supposed to use the letter with an arrow over it, but I can’t figure out how to do that. Please leave any tips in the comments.)
OK. I know that picture doesn’t make much sense, but here’s the skinny. Let’s say a particle traveled along vector A, then turned and traveled along vector B. How far did the particle actually go from its starting point? One might think that all we would have to do is add the magnitudes of vectors A and B, but one would be wrong.
Look what I did. I stuck the tail of vector B on the head of vector A, then drew a blue vector R from the tail of vector A to the head of vector B. That blue vector is called a vector sum or resultant. Vector R is the distance from the starting point that our little particle traveled. We can also get the direction the particle went.
Now, let’s do a sample problem that I totally ripped off from my text book. (It’s not cheating because I totally got it right.)
A hiker hikes 1.00 km north, then 2.00 km east on a flat field. How far and in what direction is the hiker from the starting point?
OK, we’ve got this. Let’s do it together. We need to find the vector sum of two vectors (1.00 km north and 2.00 km east,) as well as the direction the hiker traveled, aka the angle Φ. This is just like the example above. We need to draw this out. Don’t have a pen and paper handy? Don’t worry. I drew it for you.
You’ll notice that I attempted to draw the diagram to scale. That way all I have to do is measure it. The answers I came up with were a displacement vector R about 2.25 km and about a 64 degree angle Φ. But that isn’t very precise. We. Need. Math.
Because the given vectors create a right angle, this should be easy peasey lemon squeezey. All we need is a pinch of algebra and a dash of trigonometry.
You’ll notice that the given vectors create a right angle. Vector R is the hypotenuse of the right triangle. That means all we have to do is use the Pythagorean theorem to find out length:
√(1.00 km)2 + (2.00 km)2 = 2.24 km
To find angle Φ, all we have to do is find the tangent:
tan Φ = opposite side/adjacent side = 2.00 km/1.00 km
Φ = 63.4 degrees
Wow! My manual measurements were pretty close!
I wish I could go into more vector things, but I’m running out of time. (That’ll teach me to leave this post until Sunday evening.) I think vectors are awesome, and I can’t wait to do more work with them.
Before I leave you, I want to send many, many Internet high fives to those who have tried to explain some of this physics stuff to me. Commenter GravitysWings basically wrote a whole post in the comments explaining limits and derivative rules. I don’t think I fully grasp it (in fact, I know I don’t) but I know it will come in handy as I get deeper in. Also, I’d like to thank Carl Tracy of helping me with instantaneous velocity and instantaneous acceleration via a Google Hangout, which is less than idea. What a trooper. Gold stars and high fives!
One more thing. Last week I mentioned that this class is a 12 week course. I thought I could teach myself in that amount of time. I was arrogant. Well, my darlings, I have been humbled. I think that, if I had a teacher, maybe. But I’m teaching myself, so I will likely need more time. But I will finish! That I promise you. What that means for you is that I’ll probably be blogging about physics from now until the end of the world. I hope that’s cool with you. (Also, go thank a teacher. NOW!)
So, physics people of the Teen Skepchick community: Did I get it right? Did I mess up? I know there is a lot I left out. Is there anything I need to clarify?