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I) How much tension must a rope withstand if it is used to accelerate a 1210-kg car horizontally along a frictionless surface at 1.20 m/s$^2$ ?
(I) What force is needed to accelerate a sled (mass = 55 kg) at 1.4 m/s$^2$ on horizontal frictionless ice?
(I) A 7150-kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 m/s. A 3350-kg load, initially at rest, is dropped onto the car. What will be the car's new speed?
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welcome to Section one of Unit two in this section will be considering vectors. We're looking at this first before we really dive into to Deacon O Matics. Because if we don't understand vectors than all of two dimensional Kinnah Matics is going to be very, very difficult to keep track of now again. Basics of vectors. We have direction and magnitude, direction and magnitude. This is different from scaler quantities, which only have a magnitude. We've already discussed what different types of vectors we have. We have things like position and displacement. We have velocity, we have acceleration. Meanwhile, for scale er's we had total distance and we had speed. Okay, where you can also think of speed sometimes as being the magnitude of the velocity. Now, these two absolute value science are very important to understand here. When I put these two signs down, what it means is I'm looking for the magnitude of a vector. So let's look a little bit about what a generic vector is and how we can write it down so that we can understand what all the different symbols they're going to mean. So a generic vector is generally written in one of two ways will say that we have a vector A and it's written as we have some amount a X in the eye hat direction, plus some amount a Y in the J hat direction. So when we draw a picture of this, what it means is that we have a vector A but the component it components of it are We have one piece that's in the I had direction. Horizontal with an amount, a X and one piece That's vertical in the direction J hat with an amount A y. Okay, so this is one way to write out this vector. Thea Other way to write out this vector looks a little more like polar coordinates. If I were to say I want the magnitude of the vector that's equal to the square root of a X squared plus a y squared notice. This is just the Pythagorean theorem. In order to find the size of a here. Now in order, Thio, finish this other way of writing things out. I need also an angle, an angle. So the angle I'm looking for is this one which again this should look familiar. We're going to have the inverse tangent function of a Y over a X. If you remember the video on transformations between different coordinate systems, then this should look really familiar. All we've done is we've gone from on X Y notation to one our Seda notation. Okay, so we have a magnitude and direction and angle here. In this case, we're not really so much showing magnitude is direction is how much a component view how much the vector is should pointing in the two different directions. So both of these have their different utilities. Generally speaking, when we talk about vectors, we're going to use the R theta form. We're going to say Sally kicks the ball at 10 m per second at 20 degrees above the ground. Something like that. We're not gonna report it in component form, but it turns out when we go to do mathematics, component form is the form of choice because if you try to do mathematics with the R theta form, you're gonna end up tripping all over yourself. It's going to be tough, and we'll look at how that works in the coming videos. The basic idea of it, though, is that when you go to add two vectors kind of like in our picture here. If I were to add to generic factors to each other A. We'll call it and be, and I want to know what the some of them is. Then I would draw a line like this and call it See, I'd say, a plus be sequel to see, Don't not make the mistake of thinking that the magnitude of a plus the magnitude of B is equal to the magnitude of C. It's simply not true again. We could break this into right triangles and you would be able to see that from the Pythagorean theorem. You can especially see it over here. You know that the magnitude of a here is not equal to a X plus a y. It's a squared is equal to a X squared plus a y squared. Similarly, this is not a true statement, but a plus B equals C is true. So how do we deal with that? Well, what we're gonna do is we're gonna ride out a and component form, as we had before A X I have plus a Y J hat and we're gonna write be in component form, so that will be be X. I had, plus B y j hath. And now I'm gonna add the two in component form. So if I take a plus, be here and I group terms, what I'm going to get is a X plus BX in the I had direction plus a y plus B y in the J hat direction in this would be equal to see. Then when I go to find the magnitude of C that would be equal to the square root of a X plus B X squared plus a y plus B y squared. And when I go to find the angle of C, I would say that's equal to the inverse tangent of a Y plus B y over a X plus b X. So there's the fundamentals of how to add vectors now. Obviously, if we want to add vectors, we're also going to be in situations where we want to subtract vectors. There's two ways to handle this. I'm going to show you the way that I like, because I feel like it draws a better picture. So say were given two vectors a pointed this way and B that's pointed this way. Now I should say that vectors. One thing that's odd about vectors is they don't care where they are. You can move them around freely. If I were to copy this thing and then tell it that I want to move it around, okay? That's an allowed operation with vectors. And I could put it here on the tail of B in order to add it to be Uh huh. That's a perfectly allowed operation with vectors. You can move them around. Doesn't matter where their oranges, it their origin is. As long as they maintain the same magnitude and direction, they still remain unchanged. So we have Victor and Victor B, and instead of adding them in this case, we want to subtract them. Well, let's look at what that does. If I have a minus, bi, what I'm gonna end up with is a negative B x I hat minus b Y j hat. And remember, here what the minus is doing, just like in one Deakin Matics is it's telling you the direction. It's not really telling you much about the size of the vector, but it is telling you now we're moving in the negative I hat and the negative J hat directions we want to move in The opposite direction is what we had before. So what we're gonna do is we're actually going to take this vector and flip it around. So a plus B then or rather a minus bi can be visualized as the a vector minus be or rather, plus a negative B and then see would be there so we can draw it the same as before. All we have to do is remember to multiply BU by a negative, which will flip it as a vector 180 degrees, so multiplying a vector by a negative flips it 180 degrees. That's an important thing to remember, and then when we go to combine them and we say a minus B, supposing we have the same components is before. What will have is that C is equal to a X minus. B X All in the I had direction plus a Y minus B y all in the J hat direction. So it looks exactly the same is adding we've just multiplied the components of B by negative signs. Now this is a good example of why math is much better done in component form. But it's often better to report in the artha to form so very often. What you'll find yourself doing is going back and forth between the two. If you didn't watch the conversions video, all that's different between these two things is a right triangle. You have a magnitude on the high pot noose and an angle right here. So effectively you have are is the high pot news and then you have the different components. Here you have X and why. So in this case, are is going to be equal to the square root of X squared. Plus y squared Fada will be equal to the inverse tangent of why over X X is going to be equal to r cosine theta and why is going to be equal to our sine theta? And we'll do this over and over again in the coming videos
Newton's Laws of Motion
Applying Newton's Laws