Vectors and Projectiles Legacy Problem #19 Guided Solution

This problem falls into a class of problems which are commonly referred to as riverboat problems. Before I actually discuss the problem, I'd like to call your attention to a webpage that's linked to here from the solution page. The webpage is titled Relative Velocity in Riverboat Problems, and it's a good read, particularly if you're having difficulty with these types of problems. Now, here we have Glyndon and Harold, and they're trying to cross a river. We're told the river flows east. So if you think about river, a river flowing east means that its banks stretch from east to west, and so horizontally on a page of paper, and Glyndon and Harold are traveling north. So what I'm going to do is I'm going to diagram the situation. I wouldn't think of doing this problem without diagramming the situation. So I begin by drawing two horizontal lines representing the northern and southern banks of this river, and I draw Harold heading towards the east, representing the direction the river flows. I can label that arrow as river. And then Harold and Glyndon are traveling north, so they must be starting on the bottom line, the southernmost line, and heading up north. So I draw an arrow representing their velocity of 2.4 meters per second. In fact, I label it 2.4, or maybe even V-bolt equal 2.4. It's a northerly directed arrow, and it has an arrowhead on it. Where that arrowhead ends, I'm going to draw another arrow going east and label it V-river, and it equals 1.9 meters per second. Now I'm adding two vectors so that I can answer part of this problem. Find the resultant velocity of the bolt, both magnitude and direction. So to do that, I'm going to draw the resultant velocity. It's the resultant of two added vectors, one being north, the other being east, one being magnitude 2.4 north, the other 1.9 meters per second east. And the resultant is simply found by using the Pythagorean theorem that gets you the magnitude. And to find the direction of that vector, you need to first draw it and recognize that it makes some sort of angle with a 2.4 meters per second north. And label the angle back at the tail of the vector. It's so many degrees east of that north, label it theta, and theta is equal to the inverse tangent of the opposite side, 1.9, to the adjacent side, 2.4. Theta equals the inverse tangent of 1.9 divided by 2.4. Find out what theta is. I get 38 degrees when I find it. That's the number of degrees rotation east of north. If you drew your triangle differently, you might be finding the number of degrees rotation north of east, or counterclockwise from east. Take a deep breath. You're ready to do two more problems, and these remaining two problems have to do with the equation d equals b times t. In the first problem, b, we wish to determine the time to cross the river. Now, when we speak of the time to cross the river, what we mean is the time to head north across the river. Even though we're not really heading north, we're heading north and east, we can use the northerly speed and the northerly distance to find that time. The northerly distance is 38 meters, after all. That's how wide the river is, and when they measure width, they measure it directly straight across from bank to bank. That's a 38 meters north, so what I'm going to do is I'm going to say d equals b times t, where the d is 38, and the b is 2.4. Not the 1.9 and not the 3.06, but the 2.4, because that's the northerly speed and I have the northerly distance. You go over time, and you get something around 15.8 blah blah blah seconds, and now you're going to use that in part c, because now you want to find a easterly distance. How far down the river, how far east do Glenda and Harold travel when they reach the opposite shore? Well, they're carried that direction by this 1.9 meters per second flow of the river, and so to find that distance, I'm going to go d equals b times t. I'm going to use the same t I just calculated, but I'm going to use 1.9 meters per second as my b. It comes out to be about 30.1 meters. You can round it to 30 meters if you wish.