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In reaching her destination, a backpacker walks with an average velocity of 1.34 $\mathrm{m} / \mathrm{s}$ , due west. This average velocity results because she hikes for 6.44 $\mathrm{km}$ with an average velocity of $2.68 \mathrm{m} / \mathrm{s},$ due west, turns around, and hikes with an average velocity of $0.447 \mathrm{m} / \mathrm{s},$ due east. How far east did she walk?

805$m$ to east

Physics 101 Mechanics

Chapter 2

Kinematics in One Dimension

Motion Along a Straight Line

University of Michigan - Ann Arbor

Simon Fraser University

McMaster University

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all right. So problem 10 is kind of a deceptive problem and that it doesn't read like a particularly challenging one. But there's a lot of stuff that comes out of the woodwork for this one. So with that, let's get started. Now. This problem kind of starts by throwing a ton of information at you about people Ah, hiker going westward and then turning around and going eastward before we start kind of dissecting it and pulling out the numbers. I want to take a second to define what we mean by west versus East. So west and east are opposite directions. And in physics, that means one of them is positive and one of them is negative. Now, I'm using kind of these standard notation, which is that East is always gonna be positive. West is always going to be negative languages again, kind of a standard notation. If you switched it around it the other way, you would still end up with the correct answer like it does not affect your final outcome. It's just kind of telling us which way is which. So the pieces that were given here to work with is first up the total velocity of this hiker is 1.34 meters per second to the west and because West is the negative value, I'm going to call that negative 1.34 meters per second now. The next piece we know is that the first part of the trip, which is the westward part of the trip she was hiking with 2.68 meters per second again because this is the westward direction, it's going to be a negative value than the direction traveled to the west is 6.44 kilometers now. We won't be working in our base units, so instead of 6.44 kilometers, I'm going to change that into 6440 meters and since that's also in the west, we have negative 6440 meters. The last piece of information were given is that the velocity in the second part when this hiker is traveling to the east is 0.447 meters per second and that's it. In terms of the information we're given now, our goal here is to figure out how far she travels to the East what this x e value actually comes out to be. So the equation we're gonna be using for this and we actually do just use one equation we just use it kind of creatively is the idea that velocity is equal to a displacement divided by the time that displacement takes. And if we look at the first part where we're talking about it going to the West, we actually know two of those values already so we can find the time of that travel. I take a second and rearrange that we have Time is equal to your displacement, divided by the velocity. And if I plug in my two values here, so I have time equals negative. 6440 divided by the velocity of negative 2.68 we get a time of 2402 0.9 seconds. So for this problem, we know everything about the first part, and we know what the end result is going to be. But we don't know our displacement or the time spent traveling to the east. Now, this is where things start getting a little bit exciting, so I'm gonna take a second here and tidy up my workspace, um, of these values. And I'm gonna move this up into the corner so we still have it available for reference. Okay, So in order to solve this problem, we have two unknowns. We have the X e value and the T E value. So my goal is to find two equations that use those two values so we can combine those equations and salt for them. Now, the first equation I want to think about what I'm gonna call equation one were Yuzhin. One is the fact that we know what the total velocity is equal to. We have a number for that. And since velocity is equal to the displacement divided by the time than the total velocity is equal to your total displacement divided by your total time. So my total displacement is my w part plus the east part and same deal for the total time we have the time spent traveling to the West added to the time spent traveling to the east. Now I can go a little bit more in depth of this because I actually have a good number of good. A few of these variables already solved for So instead of acts, W I have negative 6000 440. I don't know what X is. And on the bottom. I know the time spent to the West is 2000 402.9. But I don't know what the time in the East is, and that is equal to my total velocity of negative 1.34 meters per second. Okay, so that's one equation. It has both of our unknowns in it. It has the XY and the T E value, but no other unknowns. So I'm gonna store that over here, remember, from algebra class, in order to solve a series of equations system of equations, you need a number of equations equal to your number of unknowns. So I have two unknowns and one equation, which means there is going to be one more equation that we need to find. I'm going to call equation, too. Now. We already know everything about Part one, so that's not too helpful. However, the velocity in the east is equal to our displacement in the East, divided by the time in the east. Now, when I plug in my values. Here we have 0.447 meters per second for that velocity to the east is equal to the displacement in the East, divided by the time spent traveling to the east. And I'm not gonna multiply by t e. So I have X e. The displacement in the east is equal to 0.447 multiplied by the times been traveling to the East and that is my second equation. So I now have two equations that are solving for the same two pieces. We have the displacement in the East and we have the time in the East. So I'm going to use the substitution principle here and plug equation to into equation one. And this is where the long math starts. So let's get to it. We have negative six negative 6440 plus instead of x e. I'm gonna take my value from equation to and plug that in. So I have plus 0.447 times the time spent traveling in the East. That whole piece gets divided by 2402 0.9, plus the time spent traveling to the east, and that whole mess should be equal to negative 1.34 meters per second. So now it's just a math problem where we're simplifying and solving for that T e value. So my next line is negative. 6440 plus 0.447 time spent traveling to the East is equal to negative. 1.34 multiplied by 2402.9. Plus the time's been traveling to the East. Then I'm going to distribute that value out. So my left hand style side stays the same for a minute or 70. But the right hand side changes to be negative. 3219.9 plus, um, a negative 1.34 t. Okay, so from here, we're going to add our negatives to each side. So we're grouping like terms, and we end up with 1.787 multiplied by the time spent. Traveling in the East is equal to 3220.1, and then I divide by 1.787 to get my time spent traveling in the east of 1801.96 seconds. So that is a long drawn out their vision to get down to what the time spent traveling in the East is actually going to give us. That's the bulk of the work here. Now our end goal is actually figured out that X value the displacement in the east. But we can take this value and really quickly plug it into equation, too. So we end up with on X E value equal to 0.447 multiplied by 1801.96 which gives us our final answer of 805.5 meters. And that is a positive 805.5 meters because, as we established way back start of the problem, positive is the eastward direction.

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