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Race Against Time

Patrick Ford
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Hey, guys, let's work out this problem together. We're told that a runners trying to complete a race, they're running at some speed for some time. Then they realize they have to go faster. And we have to figure out how fast they have to go in the second part of the race so that they finish in some time. So this is a classic motion with multiple parts. So let's just stick the steps, draw the diagram and list out all the variables. We know that from start to finish, the entire race is going to be 100 m in 20 seconds. But there is a point where they realize they have to go faster. So there is a point right here where they're going to change speeds. So remember, we have to label all the intervals. This is gonna be a B and C. And so I'm gonna list out all the variables for this one in red and this one in blue. So list out the variables we know we're gonna have Delta X from A to B. We're gonna have velocity from A to B and T from A to B. Also noticed that in the problem. It says that we're in constant or average velocity for a whole entire time, so there's really only just those three variables to keep track of now. For the second part, we have Delta X from B to C. We have the velocity from whoops. That's the velocity from B to C and then the time from B C. And now finally, we remember we have the whole entire thing. There's two parts. There's also a whole interval that we have to keep track of. That's this guy over here. And we know that this is the total. We know that the total distance for this race is 100 m, and we also know that the time that we're supposed to complete this race is 20 seconds. So we know what both of those values are for Delta X and T. So let's just go through now and identify the other things we know about the problem. In the first part, we're gonna be running for 4 m per second for 14 seconds. So that means that the velocity from A to B is gonna be four. And the time is gonna be 14. But we don't know what the distance is in the second part. We actually don't know anything about this problem. We don't know anything about this interval. We don't know that the distance, the velocity or the time. In fact, the velocity in the second part is actually what we're trying to figure out. What's the average velocity for the rest of the race to complete it in 20 seconds. So this is gonna be our target Variable over here. So now that we've drawn the diagram and we've listed all the variables now we just have to go ahead and solved by writing the equations for each of the intervals. So remember, we're working with constant velocity. There's only one equation in each interval that we consult for. So the velocity from B to C is gonna be Delta X BTC over T B to C. So, basically, all we have to do is just figure out. Remember, Delta X B C is gonna represent the distance left at the end for the end of the race, and the TBC is the time that we need to finish it toe in order complete under seconds. So I just have to go ahead and figure out what each one of these variables are in order to get that that that average velocity that I need. So whenever I'm stuck and I need to figure out what Delta X B C is, um, there's only one other equation that I could go and find. So if I don't have it over here, then I have to relate it to the total the total distance, which is Delta X from A to C. I know that this distance is 100 m, but I also know that this distance is made up of both of these two distances added together. So basically I can set up in equation Delta X from A to B plus Delta X from B to C. And now I just have to figure out if I actually know two of those things. Obviously, I know the total distance is 100 but if you look through this problem, I don't know what either of these things are. And this is really what I'm trying to look for so I can plug it back into this equation. So can I figure out what this delta X from A to B is That's the question. So can I figure out what this thing is? Well, if I go over to this interval over here, I have two out of the three variables that I need so I can actually solve for Delta X from A to B. So we start off the equation V A to B is equal to Delta X from A to B over ta to be. And so I could rearrange for this really quickly on dykan get that Delta X from A to B is equal to V A B t A B, and I actually know what both of these things are. So I could just go ahead and solve. This is gonna be four times 14 which is gonna be 56. So now I can just pull this back into this formula and figure out what that distances. So I know that Delta let me set up this a little bit differently. I have 56 plus Delta X B C is equal to 100 right? The first part from the second part is plus the second part is equal to 100. So that means that Delta X from B to C is 100 minus which is equal to 44 m. So that's 44 m, which I've got right there. So that Z, that's the first variable that I that I need and the second one that I need now is the time. So remember that there's two. There's two variables that I was missing in this formula. I also need to figure out the time Well, if we go, if we go figure out the time over here the time the first part was seconds. But the time that I need to complete the race in is 20. So this is pretty simple, but basically the time that this needs to be is six. Because I have to finish the race in 20 seconds. So if this is 14 and this has to be six, so now I have everything I need. So the V B C is equal to 44 and then divided by six. And if you work this out, you're gonna get 7.3 m per second. So they have to speed up to the speed in order to finish the race in that time. Alright, guys, that's over. This one. Let me know if you have any questions