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Vectors
In chapter one, I mentioned that vector quantities such as displacement, velocity, and acceleration have to have a direction. This chapter will now go in more depth about what that entails. All vectors can be represented by an arrow that is pointing in a certain direction. The arrow itself gives you two pieces of information. One, it tells you the size of the quantity which we call the magnitude. Two, it tells you which direction the quantity is heading. For example, in the diagram below, there is a vector arrow that represents velocity. The actual length of the arrow is important because it represents the speed of the vector. In this case, if 1 cm = 1 m/s, than the speed of the object would be 5 m/s, and since it is pointing East, you know that the object is traveling 1 m/s East.  If you wanted to represent a velocity of 10 m/s West, you would make an arrow that is twice the length of the one above and have it point West. As said before, displacement, velocity, and acceleration can all be represented by an arrow. Adding Vectors Adding two or more vectors together is not like adding two numbers together. The direction has to be considered. There are two steps to adding vectors: 1. Take the tail of vector two and place it on the head of vector one. If you have more than two vectors, take the third vector's tail and place it on the head of vector two and repeat until you run out of vectors to add. 2. To find the resultant, draw an arrow that starts at the tail of vector one and ends on the head of the last vector added. In the case of two vectors, it would stop at the head of vector two. When using the two steps above, a few things need to be considered. You can move vectors anywhere on the page as long the arrow itself is the same length and pointing in the same direction. That is how you take vector two and place it on vector one. The head of the vector is where the pointy part of the arrow is located, and the tail is the opposite end. The resultant is just that; the result of adding the vectors.  Consider two displacement vectors that lie along the same line as shown in the diagram below. In order to add them, you apply the two steps outlined above. It does not matter which vector you make one and which vector you make two. The resultant will come out the same.  Or, you can have one of the vectors pointing in the opposite direction.  You may have noticed that the resultants on the last two examples have been found by simply adding or subtracting the numbers. This is true. Whenever, the vectors lie along the same line, they can be added or subtracted depending on their directions. However, when they are in different directions, that is not the case. Consider the next example.  You cannot find the resultant by simply adding 3 and 4. In this case, the resultant becomes the hypotenuse of a right triangle.  In order to find the resultant, you have to utilize the Pythagorean Theorem where,  So, solving for the resultant yields,  Which means the resultant equals 5 meters. Now that takes care of the resultant's magnitude. You now have to find the direction. This is where the angle comes in. The funny looking symbol inside the triangle is the greek letter theta. That symbol represents the angle. In order to find that, you need to use the trig function tangent. Usually when you use the tangent function, you know the angle and one of the sides of the triangle, and you use the function to find the other side because the tangent of an angle equals the opposite side of the triangle divided by the adjacent side. In this example that would be 4 divided by 3. However, we don't need to solve for one the sides because we know both of them; we need the angle. This is where you use the inverse tangent give by:  The opp means opposite side of the triangle to the angle, and adj means the adjacent side to the angle. Using this function will give the angle of the triangle. In this example, the inverse tangent of 4/3 equals 53 degrees. Therefore, the vector diagram can be written as,  and, it can be written out as 5 meters 53 degrees East of North. Subtracting Vectors Subtracting vectors actually ends up being exactly like adding vectors. Let's say that you wanted to subtract the following velocity vectors.  You can express this vector subtraction in equation form as,  Where V is the resultant. Now it looks like you can just subtract the numbers right? That is what the funny little arrows on the tops of the v's take care of. Those arrows mean that these are vector quantities. So, you cannot just subtract the numbers, you have to consider the direction as well. Now, if the vectors are along the same line, then you can treat that as a normal equation and subtract the numbers. Back to subtraction. In order to handle this, you have to take the second vector in the equation and do what is called add the opposite of that vector. What that means is take the second vector and flip it so that it is going the opposite direction. Now, you can simply add it to the first vector. By the way, it is very important here that each vector is used in the proper order. You just can't take V1 and flip it over and add it to V2. You have to flip over the vector that you are subtracting. Anyway, the vector diagram now looks like this.  You are now back to just adding vectors like before. From here on out, whenever adding or subtracting vectors must be done, I will just give the answers and not go through the work necessary to find the resultant or the angle. If you have any problems, refer back to this section on how to find those items. Also, the questions in the archive page under chapter two address adding or subtracting vectors with the explanation providing the details on how to find the resultant and angle for that particular problem. Relative Velocity We can now take adding and subtracting vectors and apply it to real life situations. Relative velocity is all about which reference frame you are in. For example, picture yourself at an airport where they have those moving sidewalks, and you are standing off to the side observing people go by on the moving sidewalk. (you are not on the sidewalk). Let's say that the sidewalk is moving at 2.0 m/s to the right. If a person is on the sidewalk and is at rest relative to the sidewalk, which is the sidewalk's frame of reference, how fast is that person relative to you? If you said 2.0 m/s don't give yourself a cookie because that was not very hard. Now let's say that the person is walking 2.0 m/s to the right, and there is another person at rest on the moving sidewalk. (There are now three people: you, the person walking on the sidewalk, and the person standing still on the sidewalk). What is the speed of the person walking relative to you? I hope you said 4.0 m/s. It is the speed of the sidewalk plus the speed of the person because they are both going in the same direction. However, the speed of the walking person relative to the person standing on the sidewalk is 2.0 m/s because they are both going 2.0 m/s to the right due to the sidewalk. So, the person that was walking on the sidewalk had two different speeds because he was observed by two different people in two different reference frames. We can extend this to two vehicles traveling. Let's say that you are in a truck traveling at 30 mph to the right, and there is a car behind you traveling at 40 mph. We can find the velocity of the car relative to the truck. That means that we will find the speed of the car as seen by you in the truck. You can actually figure that out without any equations because if the car is traveling 10 mph faster than you are, then the velocity of the car as seen by you will be a + 10 mph if we make right the positive direction. (The positive means the car is coming at you).  Here is the equation to find that answer.  Here is what those terms mean. The Vct is the velocity of the car relative to the truck, which means it is the velocity of the car as seen by the person in the truck. The Vc is the velocity of the car, and the Vt is the velocity of the truck. Since this is subtracting vectors, you have to add the opposite of Vt. Mathematically, since Vt is a positive 30 mph, because it is going right, then the opposite of that would be negative 30 mph. Since both vectors are along the same line, you can simply put the numbers in and you get, I put the -30 in paranthesis to show adding the opposite of Vt. This means that the car looks like it is going 10 mph towards you because that is the velocity of the car from your reference frame. The cool thing about this is the car would look no different to you if you were at rest and the car was traveling 10 mph toward you. The two situations are identical from your frame of reference.  You can also show this result by using the vector arrows as shown below.  How about an example where the car is traveling + 30 mph and the truck is traveling at +40 mph? The equation becomes, The negative answer means that the car looks like it is now going 10 mph away from you. Now let's make the car be traveling at 30 mph South and the truck be traveling at 40 mph East. Now the two vectors do not lie along the same line so you cannot just plug in the numbers. You have to set up the vector triangle.  Here, the vectors are given by,  Since you add the opposite of Vt, you have,  Which yields,  Therefore, the velocity of the car relative to the truck is 50 mph 53 degrees W of S. So, if you are in the truck watching the car, it looks like the car is heading in the direction of the resultant vector at 50 mph. One of the cool things about relative velocity is that without outside stationary objects, you could never tell whether or not you were moving and the other vehicle was at rest or you were at rest and the other vehicle was moving. (More on that in chapter three). Let me illustrate this further.  Pretend you are in the red vehicle out in the middle of space millions of miles from any planetary object. Let's further suppose that the velocity of the red vehicle as seen by you is 10 mph towards you. Also, both of you are traveling at a constant speed, which means you do not feel that you are moving because there are no bumps in space and there is nothing to look at outside the window to tell you that you are moving. Here is the cool part. There would be no way for you to determine who was moving because these are the possibilities: 1. You are at rest and red is moving 10 mph towards you. 2. You are traveling to the right at a certain speed and red is traveling toward you with a speed that is greater than yours by 10 mph. 3. You are traveling to the left at a certain speed and red is traveling to the left with a speed that is less than yours by 10 mph. 4. You are traveling to the left at a certain speed and red is traveling to the right with a certain speed that yields a relative speed of 10 mph. 5. You are traveling to the left at 10 mph and red is at rest. The point is, all five possibilities will look exactly the same to you! There is nothing you can do inside your vehicle to determine which of the five is correct. They are all correct. Horizontal and Vertical Speed An interesting question in physics is does a horizontal speed affect the vertical speed of a freely falling object? For example, if you shot a bullet horizontally and dropped another bullet from the exact same height and the exact same time the other bullet left the barrel of the gun, which bullet would hit the ground first? The answer is surprizingly neither! They would both hit the ground at the exact same time. This is difficult to swallow at first. Almost everyone says the bullet that was shot out of the gun because it is traveling at such a high speed. This is true. However, that speed is horizontal which has nothing to do with the acceleration due to gravity. Gravity pulls straight down which causes the object to have a vertical velocity, or y-velocity. The horizontal velocity, or x-velocity, does not make gravity pull any more or less. Consider the picture below. |
