Vectors and Projectiles Legacy Problem #13 Guided Solution

This problem is going to require a little bit of understanding of football, some pretty good math skills, and the practicing of the habits of an effective problem solver. I'll help you through all of it. Here we have Avery, who's at it again, throwing a pass downfield. 36.5 yards is the distance that the pass goes, and it's 21 degrees west of south. We're going to focus here up on the football. And that's the displacement of the football from the point where it's thrown to the point where it's caught. And what we're asked to calculate is we're asked to calculate how many yards were gained on the field. So in understanding football, the way they determine the amount of yards gained on the field is they determine it as the number of yards a ball goes in the direction of the goal line, towards the goal line. Not the sidelines, the side-to-side distance, but only from end zone to end zone distance. In other words, what we wish to determine here is the component of the displacement vector in the direction of the line that goes from end zone to end zone. So what I would recommend you do, like any problem, is to begin to picture it by drawing a diagram. Actually, picture it by drawing a picture. And in your picture, you probably should start by drawing a line, a scrimmage, just a line across a sheet of paper. I'd make it a horizontal line. And then maybe a little bit above that line, not too far, put a dot. The dot represents the ball, where the ball was at when it was thrown. That's the starting location of the ball. It's a little bit above the dot, because the ball starts from behind the line of scrimmage, and our line represents the line of scrimmage. Then draw a vector that goes a little bit west of south, so you should know where south is. And a little bit to the west of it would be to the left of south. And draw that vector downfield. Put an arrowhead on it. It's going to be 21 degrees west of south. That's just a sliver west of south. Put a dot at the end of that vector as well, and that's where the ball is caught. What we wish to do is find out the component of this displacement vector in the southerly direction. And so now what I would recommend you do is go up to the first dot you drew from, that's the dot above the line of scrimmage, and draw an x-y coordinate system. And draw a horizontal line and a vertical line. Extend that vertical line as far south as the actual second dot that you have. And then what you want to do is make a right triangle, because it's all about right triangles in this unit. So in order to make that right triangle, go to the second dot, the one below the line of scrimmage where the ball was caught, and draw a horizontal line across until it intersects the vertical axis, and then again draw a vertical line from that dot upwards until it crosses the horizontal axis. And then somewhere on there you want to see a right triangle, actually two of them. And you're going to focus on the right triangle that has its vertical side, the southerly axis, and then as a horizontal side, it just simply has that horizontal line that you drew in there. Now, inside of that triangle, up by the origin, there is a 21 degree angle. And the hypotenuse of this right triangle is 36.5 yards. And what you're going to do is you're going to take the 36.5 yards and the 21 degrees, and you're going to find the southerly side of that right triangle. The southerly side is the side adjacent to the 21 degrees, if you're looking on your diagram. And since we know the hypotenuse and the side adjacent, we have to think SOH CAH TOA, we have to think we have to use the cosine vector in order to get that cylindrical component. So we set up a cosine equation that looks something like this. We say cosine of 21 degrees equals S, where S is a southerly component, divided by 36.5 yards. We solve for X there. That's going to end up being 36.5 times cosine sine of 21 degrees. Once we get that southerly component, the final thing we have to do is subtract 7.2 yards from it because the ball did not start at the line of scrimmage. And in football, they measure the yards gained from the line of scrimmage to where the ball ended up. So you have to subtract the 7.2 yards from your answer that you got out of 36.5 cosine 21.