Problem

The three-toed sloth is the slowest-moving land m…

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Problem 6 Medium Difficulty

One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 1.50 $\mathrm{km} .$ They start at the west side of the lake and head due south to begin with. (a) What is the distance they travel? (b) What are the magnitude and direction (relative to due east)
of the couple's displacement?

Answer

(a) $\frac{9}{4} \pi=7.07 \mathrm{~km}$
(b) $45^{\circ}$

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Physics 101 Mechanics

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Video Transcript

all right. This problem has this finding the distance a couple walks as they travel 3/4 around a lake, which has a radius of 1.5 kilometers. Now, whenever you're doing a problem that involves a circle, the radius of that circle is a very important quantity. Because any equation we're going to work with, I will need the radius of the circle to be used. So this couple is starting west and walking all the way around to north 3/4 of the way around the lake. Now, had they walked the entire length of the leg, the distance that they traveled is what part is looking for here would have been the circumference of the circle or two pi r. Now thing here is they're not walking the whole thing. They're walking 3/4 that distance, so I want to take that circumference and multiply it by 3/4 to get just 75% of the circumference of the circle. Now, when I played my numbers and I have two high times 1.5 times quantity of 3/4, and when I continue to reduce and simplify this, we end up with 9/4 pie, which is our specific answer. Or if we round it, we get 7.7 kilometers. Now that's the distance they travel as they go all the way around the leg. B is looking for the displacement as they go all the way around the lake. I'm gonna shrink a down and put it over here. Um, s O displacement is different than distance because it's looking at how far away from your starting point are you? And the fact that we walked 7.7 kilometers doesn't matter because we're getting closer and closer to being back where we started. So be the displacement is asking us to find how far away this point right here. Our starting point, which actually should be on the edge of the circle, is from this point right here where we ended. Now this really comes down to looking at how far away these two points are, which we could do using some of our algebra skills. So if I connect this to the center of the circle, we know that Hey, this lake has a radius of 1.5 kilometers and going to the north side is also 1.5 kilometers. Because these air exactly on the north and west cardinal directions this is a right angle because north and west are perpendicular to each other. Now, what that means is we can treat this whole shape as a right triangle for figuring out how far away they are. And we get this nice side here now, whether you're using the Pythagorean theorem to finish up here or you remember some of the special triangles from geometry class which honestly, was never my strong suit. To begin with, you'll end up with the same idea. 1.5 squared plus 1.5 squared equals C squared that see that final value that displacement value ends up being a 1.5 times the square root of two kilometers. And this problem asked for both a magnitude and a direction. So our direction if we look at that triangle since both the X component of the triangle here and the why component of the triangle are the same length, that means our angle here. This angle above the east is going to be, um, 45 degrees above East

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