Subway Train

by Patrick Ford
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Hey, guys, let's check out this problem together. We've got a subway train and it's gonna start from rest and accelerates. Then it's gonna travel a constant speed for some time. And then finally, it's gonna slow down until it stops to the next station. So there's a bunch of moving parts here with multiple accelerations. So we're gonna solve this like any of the problem with multiple parts with accelerations. First, when you have to do first is just draw the diagram and list all the variables for all of the intervals that we have. So the first part, which is where the train is accelerating, ah, called Adobe. Then it's gonna travel a constant speed. So from B to C, it's gonna be constant speed. So this is constant. And then finally, it's going to stop I'll as it approaches the next station. So this is actually gonna be from C to D, and it's gonna be stopping here. So we've got a bunch of variables to keep track of. Let's just go ahead and list them out. So from a to B, we've got Delta X from A to B, which we know which we actually don't know we don't know Delta X me to be That's the distance In the first part we've got the velocity A and Velocity final So we know that this thing is gonna start from rest So this is just gonna be zero and let's see s We know this is zero We don't know the final velocity but we do know the acceleration of the time So I do know my acceleration from A to B and T from A to B This is just equal to 1.5 and this is gonna be 14 seconds. So now we just have to figure out whether the 1.5 is positive or negative. Well, it's going to start from rest, and it's gonna accelerates, which means it's gonna speed up in the right direction. So that's gonna be positive. This is gonna be 1.5 positive. Okay, for the second part now we've got constant speed, so constant speed for third for 60 seconds. So we've got Delta X from B to C. We don't know what that is. VB. We don't know what that easy either, because we don't know what it is from the first part v c don't know what that is. Acceleration from B to C is gonna be well, it says a constant speed. So this constant speed here actually means the acceleration is equal to zero for this part. But we do also know that the time for part B, it's 60 seconds. So whatever this velocity ends up being VB in B. C, they're actually gonna be the same thing because there's no acceleration. All right, so finally, now we have this stopping part. So we have Delta X from CDD. We don't know what that is. V c don't know what that is. V d. That's just gonna be when it stops the next station right here. So this is gonna be zero. And then finally, the acceleration from C to D Well, it's going to slow down at 3.5 m per second squared, so first accelerated. Then it went to constant speed. And then it's going to slow down, which means that this acceleration is gonna be negative 3.5, and then we got T from C to D, and we don't know what that is either. So we've got all these variables here. That's the first step. Now we just have to figure out what the target variable is in our problem. We're trying to figure what's the total distance that the train covers. So I remember we have all of the different parts of this of this motion here, but we also have the whole entire interval and this whole entire interval here there is a distance. Delta X from A to D, right. It's the total distance that you take from here all the way out to here. That's what we're trying to find. So Delta X from a D. D. Is our target variable. How do we figure that out? Well, we don't have an equation for Delta X, A two D using our motion equations, but we can't figure it out by piecing together all of the different distances in each of the parts. So right, if this was 10 2030 you would just add them all together. That would be your total distance. So this guy is actually gonna be Delta X from A to B plus Delta X from B to C plus Delta X from C to D. So we're gonna have a bunch of parts to figure out. And once we figure out all of those parts, we can just add them together for a grand total. So we've got to figure out what all of these parts are because we actually don't know what any of them are off the bat. So let's figure that out. So let's see. I'm gonna look for Delta X from A to B. So this is gonna be my target variable in this piece right here. So that's the second step, which is where we identify the known and the target variables. Now we have to pick a, um, equation, one of our motion equations that doesn't have the ignored variable notice how I have three out of five variables 123 and which means the one I'm ignoring is actually gonna be the final velocity, my VB. So that means I just need to pick an equation that does not have this vb in it. And if you go ahead and look for that, that's actually gonna be well, let's see, we have VB here, this final velocity Final velocity and this is also gonna be final velocity. So we're actually gonna pick equation number three So this is going to say that my Delta X from A to B which is actually my target variable, which is great because this is gonna be on the left side of the equation is gonna be my initial velocity times time. Well, we know this is gonna be zero, because our va is equal to zero plus one half acceleration, which is 1.5 times t squared. So this is gonna be my 14 seconds squared, so I could just actually plugged that straight into my calculator on my delta X from A to B is gonna be 147 m. So that's one of my variables. 1 47. So I'm one step closer to figuring out the total distance, so that's good. Now I just need to figure out the next part, which is the displacement from B to C. So now this is gonna be my target variable here, Delta X from B to C So we'll go and focus on that. So how do I figure this out? Well, my Delta X from B to C So which equation is not gonna use will remember that we only use the four, um, equations whenever the acceleration is constant. And it's not zero when the acceleration is equal to zero that all of our equation simplify. And we only just use one, which is V equals Delta X over Delta t So we're trying to figure out by Delta X from B u c We're just gonna use the constant velocity formulas. My V B to C is equal to Delta X B two C over T B to C, which means that Delta X from B to C is just gonna be my velocity times the time now I have the velocity or sorry, I have the time, which is 60 seconds here. So I have That's all I have to do is just figure out what the velocity is from B to C and notice how it's actually gonna be the same velocity. Um, it's gonna be either VB or B C because it's gonna be a constant velocity, right? So whatever I figured out for VB is gonna be the same thing for for B. C. Now we just have to go and figure that out. How do we do that? Well, notice how If I don't have either of them. I'm gonna have to go to a different interval and figure that out. Which other interval also has VB? Well, these either. Actually, there's only one choice. It's gonna be the first part right here. So what we could do is we can actually use the first interval because now we know four of the five variables to figure out what my velocity is that I ignored in the first part. So that's what we're gonna go ahead and do. So now what I can do is I could go back over here, and I could say, Well, now I want to figure out what my velocity B is. So how do I do that? Well, now, I'm gonna use my three out of five variables for this part here. And if I'm trying to figure the final velocity, I could just use the first equation. That's the easiest one. So I'm gonna use equation number one to figure out the final velocity for VB is equal to V A plus a times t. So I know I'm going from rest. So this is zero, and then my acceleration is 1.5 and my time is 14. So I get 21 m per second. So that means that I know that this is 21 this is also 21 which means now I could go ahead and use my formulas. Now I have the velocity and I have the time, which means I can figure out Delta X. So this history is gonna be 21 times my t B C, which is 60. If you work this out, you're gonna get 12. 60. So notice how sometimes you have to stop and you're gonna have to go to other parts in order to figure out some variables over there and bring it back to the equation that you're trying to look for. So that's the second piece you think about, like, pieces of a puzzle. I've got the second piece of the puzzle 12. 60. So now I just have to figure the last one. The last one is Delta X from C to D, So I gotta figure out this guy over here Notice how also, I figured out another variable when I did this. So remember how V c was an unknown. Well, we just figure out that V. C was 21 here, traveled a 21 m per second. So that means it ended right here and now it's 21 m per second over here and then eventually is going to stop. So now we're looking for the Delta X from C to D. We just go ahead and look at our, um, equations. We know there is an acceleration here, so we are going to use these variables. Which means we do need three out of five. I've got three out of 5. 21 0 and negative 3.5, which means that my t is gonna be my ignored variable over here. And so now, which equation we're gonna use? Well, the one that t is ignored in is gonna be equation number two. So this is when we're using for two for for a Delta X from C. D. So this is just saying that my final velocity V D squared is equal to my initial velocity V C squared, plus two times a from C d d times Delta X from C to D. So this is actually what I'm looking for here My Delta X. So I just work everything else out, you plug everything else in. So we know my V d is gonna be zero. My V c is gonna be 21 and then two times the acceleration. Well, the acceleration is gonna be negative. 3.5. Remember the negative sign and then times Delta X from CDD. So now if I square this, I want to actually get and move to the other side. If I square this move to the other side, you're gonna get negative. 4. 41 equals and then you multiply these things out. You're gonna get negative seven times Delta X CD. So now my Delta X M. C. Diddy is just Well, the negatives will cancel. You're just gonna get 4. 41/7, which is equal to 63 m. And so this is the final piece of the puzzle. The final displacement. So now you just go ahead here, you put them all back together again. So 63 I can say that Delta X from a D. D. The total distance traveled of all of the parts is gonna be 1 47 plus 12 plus 63. You add all these things together and What you get is you get 14 70 m and that's your final answer. Alright, guys, So again, a lot of moving parts. That's why it's really important to keep organized in these problems because otherwise you're just gonna get totally lost when you're trying to figure this out. Anyway, let me know if you have any questions. That's it this way.