AP Physics 1 review of 2D motion and vectors

- [Instructor] What do vector components mean? Vector components are a way of breaking any vector into two perpendicular pieces. For convenience we typically choose these pieces to lie along the X and Y directions. In other words to find the vertical component of this total vector knowing this angle, since this vertical component is opposite to this angle, we could write the vertical component as the magnitude of the total vector times sine of that angle. And since this horizontal component is adjacent to that angle, we could write the horizontal component as the magnitude of the total vector times cosine of that angle. And if instead we were given this angle and we wanted to determine the vertical component of the total vector, since this vertical component is now adjacent to this angle we'd write the vertical component as the magnitude of the total vector times cosine of this angle. And since the horizontal component is now opposite to this angle we'd write the horizontal component as the magnitude of the total vector times sine of this angle. So remember, to find the opposite side you use sine, and to find the adjacent side, you use cosine. So what would an example problem involving vector components look like? Say you had this question and you wanted to determine the X and Y components of this velocity vector. Since the horizontal component is adjacent to the angle that we're given, we're gonna write the horizontal component as the magnitude of the total vector 20 meters per second times cosine of the angle, which gives us 10 meters per second. And since the vertical component is opposite to this angle, we can write the magnitude of the vertical component as 20 meters per second times sine of the angle which gives us 17.3 meters per second. But since this vertical component is directed downward, technically this vertical component would be negative 17.3 meters per second. So using sine and cosine will give you the magnitude of the components, but you have to add the negative signs accordingly. So if the vector points right, the horizontal component will be positive. If the vector points left, the horizontal component will be negative. If the vector points up, the vertical component will be positive. And if the vector points down, the vertical component will be negative. What does tail to tip or head to tail vector addition mean? This is a graphical way to add or subtract vectors from each other. And the way it works is by taking the tail of the next vector and placing it at the tip or the head of the previous vector. And once you're done doing this for all your vectors, you draw the total vector from the first tail to the last head. In other words, if you were adding up vectors A, B, and C, I would place the tail of vector B to the head of vector A, and then I'd place the tail of vector C to the head of vector B. And once I'm done I would draw the total vector going from the first tail to the last head. And that total vector would represent the vector sum of all three vectors. And if you had to subtract the vector, you could still use vector addition. Simply add the negative of that vector. In other words, if you had some vector B and you wanted to subtract vector A, instead of thinking of it as subtracting vector A, think of it as adding negative vector A. And the way you find vector negative A is by taking vector A and simply placing the arrow head on the other end of the vector. So what would an example involving tail to tip vector addition look like? Imagine we have this question, and we have these four vectors, and we were asked to determine what direction is the sum off all of those vectors? So we'll use tail to tip vector addition. I'll take vector A preserving its direction. I'm not allowed to rotate it or change its size. And I'll add to that vector B. The way I do that is putting the tail of vector B to the tip of vector A, and we add vector C. And the way we do that is put the tail of vector C to the tip of vector B. And finally we'll add the vector D by putting the tail of vector D to the tip of vector C. And now that we've drawn all of our vectors, our total vector will go from the tail of the first vector, to the tip or head of the last vector, which means this is the direction and magnitude of the total vector, A plus B plus C plus D. Another more mathematical way of adding vectors is by simply adding up their individual components. So to find the total vector A plus B, instead of graphing them tail to tip, you can find the horizontal and vertical components separately by adding up the individual components. In other words, to get the total horizontal component of the total vector A plus B, I could just add up the horizontal component of A, which is negative 20, and the horizontal component of B which is negative five to get negative 25. And to find the total vertical component of the total vector A plus B, I can simply add up the vertical component of A which is negative 15 to the vertical component of B which is 10, to get negative five. This technique lets you quickly determine what the individual components are of that total vector. And again, if you need to subtract a vector you can still add the components, except the components of a negative vector all get multiplied by negative one. In other words, if vector A has components negative 20 and negative 15, then vector negative A would have components positive 20 and positive 15. So what would an example of vector component addition look like? Let's say you had this question. You had vectors A and B with these components, and you wanted to know the components of the total vector A plus B. So to find the horizontal component of the total vector A plus B, I can add up the individual components of A and B. So the horizontal component of A is four, plus the horizontal component of B is negative one, gives me a total horizontal component of positive three. And I could do the same thing for the vertical component. I could add up the component of A plus the component of B, which would be five plus negative four, gives me positive one. So since my horizontal component of the total vector is positive, I know it points to the right three units. And since the vertical component of the total vector is positive, I know it points up one unit. That means my total vector A plus B points up and to the right. One unit up, and three units to the right. How do you deal with 2D kinematics problems? 2D kinemtatics are projectile problems describe objects flying through the air at angles. For these objects, if there's nothing acting on them besides gravity, their vertical acceleration is gonna be negative 9.8. And they will have no horizontal acceleration since gravity doesn't pull sideways. Also, the X and Y components behave independently. That means you'll use different equations to solve for vertical components than you will for horizontal components. Since the vertical acceleration is constant you can use the kinematic formulas to solve for quantities in the vertical direction. But you can only plug in vertical quantities into these equations. Similarly, since the acceleration is zero in the X direction you can simply use distance as a rate times time to relate the quantities in the X direction, but you should only plug in horizontal components into this equation. In other words, as a projectile is flying through the air, since there's no horizontal acceleration, the horizontal component of the velocity is gonna remain the same for the entire trip. Which means the rate at which this projectile is moving the X direction never changes. But since there is acceleration in the vertical direction, the vertical component of the velocity will get smaller and smaller until it reaches the top, and that means the total speed of the projectile is gonna decrease as well as you approach the top. And then at the top there is zero vertical component of velocity since the projectile's not moving up or down at that moment. And then on the way down the vertical component of the velocity gets more and more negative, which increases the speed of the projectile. Keep in mind during this entire trip the vertical acceleration is the same, negative 9.8 on the way up, at the top, and on the way down. The vertical acceleration never changes. So what would an example problem involving 2D kinematics look like? Let's say a meatball rolls horizontally off of a dinner table of height H, with a speed V. And we want to know how far horizontally does the meatball travel before striking the floor. So the first thing we should do is draw a diagram. So the height of the table is H. The initial speed of the meatball is V, and we want to determine how far horizontally it makes it from the edge of the table. But note this problem's symbolic. We're not given any numbers so we're gonna have to give our answer in terms of given quantities and fundamental constants. The given quantities are things like H and V, and fundamental constants are things like little G. So this quantity we want to find is the displacement in the X direction, which is gonna be the speed in the X direction times the time of flight. We know the speed in the X direction, it's gonna remain constant. So this V is gonna be the horizontal speed for the entire trip. So we can plug in V for the speed, but I don't know what the time of flight is gonna be. To get the time of flight we'll do another equation for the vertical direction. The vertical displacement is not gonna be H, it's gonna be negative H since this meatball fell downward. And the initial velocity in the Y direction is not gonna be V, it's gonna be zero, since this meatball had no vertical velocity right when it left the table. It only had horizontal velocity. The acceleration is negative 9.8, but we're gonna write that in terms of fundamental constant, so we'll write that as a negative G. This lets us solve for T. We get the square root of two H over G which we can now bring over to here to get the horizontal displacement of this meatball before it hits the ground. Something else you'll definitely have to know for the AP exam is how to graph data to a linear fit. And what I mean by that is that when you graph data it doesn't always come out linear. And when you don't get a linear graph it's hard to find the slope of that curved graph. However you can force your data to be linear if you write down the expression that gives the relationship between your data in the form of a straight line. So the form of a straight line is Y equals M X plus B. Y would be the vertical axis. X would be the horizontal axis. M is the slope, and B would be the Y-intercept. In other words, if you had the expression P equals 1/2 D squared, if you just plot P versus D you're gonna get a parabola, and that means you got problems 'cause finding the slope of a parabola is hard. But if you instead choose to plot P versus D squared, you will get a straight line because now you've required P to be your vertical axis, you've required D squared to be your horizontal axis, and the slope that's multiplying what you called X is just a constant, and that means your slope's gonna be constant. So this lets us predict what the slope would be if I plot P versus D squared since the slope is always what's multiplying my horizontal axis, the slope in this case should just be 1/2. So in other words, if you force your expression to take the form of a straight line, now only will you get a linear fit, but you can predict what the slope is by looking at everything that's multiplying what you called X. But a lot of people find this confusing and strange, so what would an example problem where you have to graph data to a linear fit look like? Say you were given this question. You repeatedly roll a sphere off a table with varying speeds V, and then you measure how far they travel delta X before they strike the floor. If the table has a fixed and known height H what could we plot to determine an experimental value for the magnitude of the acceleration due to gravity? So we repeatedly change the speed and measured how far the ball went. That lets us know these are the quantities that are varying, so these are gonna be involved in the X and Y axes. But to figure out what to plot we need to find some sort of relationship between these two variables so that we could put that relationship in the form of a straight line. Now in the 2D kinematics section right before this, we derived a formula for how far a ball goes rolling off a table in terms of the height of the table and the acceleration due to gravity. So this is the expression that relates how fast it was going to how far it went. And we need to put this in the form of a straight line so it's easiest to just make this left-hand side since it's already solved for Y. So our Y quantity on our vertical axis would just be delta X and that's okay 'cause that's one of the quantities that we're varying over here. Similarly, the other quantity that's varying is V, so I'll just call V X. That means the horizontal axis is gonna be V. I was able to do that since this was just V. If this had been V squared, I would've had to have plotted V squared on the horizontal axis. And if this was square root of V, I would have to plot square root of V on the X axis. But since it was just V, I can get away with just plotting V on my horizontal axis and now we can figure out what our slope would represent. We've got Y equals M times X. Everything that multiplies what we called X is gonna be our slope. The way it's written here the M is multiplied on the right-hand side, but it doesn't matter 'cause M times X is the same as X times M. That means this entire term, root two H over G is the slope of this graph. In other words, we're gonna get a linear fit, and the number we find for the slope is gonna equal the square root of two H over G. And since there's no added B term, there's no Y-intercept to worry about over here. So how do we actually determine the experimental value for G? We take our data, we plot them, we draw a best fit line through the data. We would use points on our line to determine the slope of this line by taking the rise over the run. That rise over run would be equal to the slope, and we know that that number's gonna equal the square root of two H over G. So if we know this number for the slope, and the table has a known and fixed value H, the only unknown is G which we can now solve for.