Vectors and Projectiles Legacy Problem #6 Guided Solution

Physics problems begin as word problems and they end as mathematical exercises, and it's in the in-between time that all the critical work gets done. And that critical work begins by reading the problem carefully to identify what's going on, to represent the motion with a diagram, to identify known information, even listed in the diagram, and to identify the unknown information, and then to plot out a strategy on how to get from known to unknown. Applied here, what we read about is Carla, who's driving along the streets. She's making three different legs of a trip. I'm going to diagram them as I read them. I've got myself a fresh sheet of paper or scratch paper. I'm just going to put a diagram on there. The first leg of the trip is 1.85 miles south. So I start by putting a big dot. I could even label it starting location or start. And then from that dot, I'm going to draw an arrow down. I'm just going to draw it some given distance down. It really doesn't matter, but I'm going to label it 1.85 miles, and that's south. Now I'm going to put an arrowhead at the end of that vector, and where that arrowhead ends, I'm going to start the next vector, which is an east vector on a map that's to the right. And it's a little bit longer than the first one, so I'm going to draw an arrow off to the right 2.43 miles to the right, or east to the right of my sheet of paper. I'm going to put again an arrowhead at the end of that vector, and then where that vector ends, I'm going to draw my third vector, and it is a north vector, 0.35 miles north. So I'm going to draw an arrow up on the page of paper, 0.35 miles north, and I'm going to put a dot there, and I'm going to label it finish location. Now I'm going to draw a vector from the starting point to the finishing point, and that's the resulted vector. I'm going to put an arrowhead on that vector, and what I want to do is I want to determine its magnitude. I might have even been asked its direction, but here's just the magnitude. Find the magnitude of that vector. Now the only way to do that is to make that vector the hypotenuse of a right triangle, and you can figure out the lengths of the hypotenuse. That vector's going south, and it's also going to the east, and the amount of east is simply 2.43 miles. Hopefully you can kind of look at your diagram and recognize that indeed, that vector's 2.43 miles east. Well, I'm going to draw the east side of that vector right up there, starting, kind of sketching it from the arrowhead of the resultant across horizontally into the north-south axes, and then I'm going to figure out how far south that is. Well, if you think about the numbers, 1.85 miles south and 0.35 miles north, if you were to add those two up, that's the same as 1.5 miles south, and so that little hypotenuse is the hypotenuse of a right triangle with sides 1.5 miles south and 2.43 miles east. Now you should be able to apply the Pythagorean theorem in order to solve for the hypotenuse. a squared plus b squared equals c squared, or 1.5 squared plus 2.43 squared equals hypotenuse squared, and determine the hypotenuse.