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In a historical movie, two knights on horseback start from rest 88.0 $\mathrm{m}$ apart and ride directly toward each other to do battle. Sir George's acceleration has a magnitude of 0.300 $\mathrm{m} / \mathrm{s}^{2}$ , while Sir Alfred's has a magnitude of 0.200 $\mathrm{m} / \mathrm{s}^{2}$ . Relative to Sir George's starting point, where do the knights collide?

$x=52.8 \mathrm{m}$

Physics 101 Mechanics

Chapter 2

Kinematics in One Dimension

Motion Along a Straight Line

Rutgers, The State University of New Jersey

University of Sheffield

McMaster University

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to Max in this problem are 88 meters apart, accelerating toward each other sums gonna draw George as a circle over here and Alfred as a circle. Over here, they're going toward each other with an initial velocity of zero for both. Be not. George is zero and we not Alfred is zero. They have different accelerations. Georgia's acceleration is 0.3 meters per second square and this is just the magnitude we know they're going toward each other. So I'm really just worried about the magnitude of the acceleration. 0.300 meters per second squared, where Alfred, the absolute value of Alfred Acceleration or the magnitude, is 0.20 point 200 meters per second squared. So we want to figure out where they meet relative to George. So kind of show this distance here as 88 meters and we know that the distance traveled by George plus the distance traveled by Alfred is equal to 88 meters. So if I look at the magnitude of the displacement traveled by George, I can use the equation. Initial velocity. Trump's initial velocity of George can t plus 1/2 a of George times t squared. So this is just our displacement formula and the initial velocity of Georgia zero. I could do the same thing for Alfred. The initial velocity of Alfred Times T plus 1/2 Alford's Acceleration Time T Square. So these two quantities, if I figure them if I set expressions for them, they need to add up to 88 meters. So starting with George, we have I know this whole term goes away in this full term goes away. So I have 1/2 tens Georgia's acceleration so 0.3 meters per second squared times T squared, plus Alfred displacement of the magnitude of Alfred's displacement, which is 1/2 10.2 times the time squared. That all has to add up to 88 meters. So this isn't directly giving us the position where they're gonna meet, But it does give us the time where they're gonna meet, and that can help us quite look so half of 0.3 point 15 So I'm just going to simplify this a little bit. 150.15 tens t squared half a point to is 0.0.1 times t squared, and that is 88 meters I can combine like terms because each of these have a T squared, so that would just be 0.25 meters per second squared can C squared and that's 88 liters. So all I need to do is divide both sides by 0.25 to get t squared and we get a T Square value of 352 seconds squared. So take the square root of both sides and we have a time of 18 0.76 seconds. Again, this isn't quite answering the question because we want to know where they need not just win any. But if we know when they meet, we can go back. And it does ask us to find the position relative to George. So if I use George's equation over here, which is the V, not G, just circle this initial equation that we can go back to right here. Me, not G 10 t plus 1/2 acceleration of George Times t squared. So we'll use that and plug in the numbers that we now have because we have the time. So the position relative to George is equal to Georgia's velocity initially, which is still zero anything. Times zero is still zero. But oh, plug it in any way plus 1/2 times 0.3 meters per second squared times 18.76 seconds and this quantity is squared. So our value for George comes out to be 52.8 meters. That's the position relative to George, where the two nights meat, so a little bit more than halfway, that's it.

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