You walk to the right at 3m/s for 8s, then turn around and walk backwards at 2m/s for some unknown time. You end up 16m to the right from where you started. How long did you walk backwards?

Hey, guys. Let's work this one out together just in case you need a little bit of help. So we're gonna be walking to the right at 3 meters per second, then we're turning around and walking backwards at 2 meters per second, and we have to figure out how long did we walk backwards for. So this is multiple parts because we're dealing with different but constant velocities over here. So let's just stick to the steps, We're just gonna draw the diagram first. So we've got the part where we're going to the right and then we're turning around and we're walking backwards. But we know that we're gonna end up 16 meters to the right from where we started. So we know this piece is gonna be a little bit shorter. So that's the diagram. Let's just go ahead and list the variables for each of the intervals. So I know that velocity from a to b by the way, it's always a good idea to label your intervals. This is gonna be b. It's like kinda like both of these things, and this is gonna be c over here. So we've got velocity from a to b, delta x from a to b, and then t from a to b. We know that the velocity is equal to 3 meters per second, and we know that the time was equal to 8 meters per second. We don't know the distance. In the second part here, we need the same thing. So we have VVC. We need, delta x from B to C. And then Tvc. So in the second part here, we know that we're walking backwards at 2 meters per second, so that means that VBC is 2. But what's important about this is that remember the velocity is a magnitude and a direction, and the direction is controlled by the sign. If we're walking backwards at 2 meters per second, then that means it's not 2, it's negative 2. It's actually really important that you remember that sign there. So our velocity is negative 2. We don't know the distance. And the time is actually what we're trying to look for. That's how long do we walk backwards for? So this is really our target variable over here. So this is what we're trying to look for. We have one other interval to consider. Remember, there's the 2 parts, but there's also the whole entire thing. So that means that that's the interval from a to c. So that's gonna be over here. And the only thing we know about this is that we end up 16 meters to the right from where we started. So that means that delta x from a to c is equal to 16, and the only other variable that we have is tac over there. Okay. So that's all of our variables and, yeah, that's the first step. Now we just have to go ahead and write the equations for each interval. So remember that in this second part here, actually for the whole problem, there's only just one equation we can use. It's the average velocity formula. So we know that the velocity from B to C is gonna be delta x from B to C over TB to C. Remember, we're looking for this time over here, so I can actually just rearrange for this equation, and I get that TBC is equal to delta xbc divided by the velocity. So I actually know what the velocity is. That's the negative 2. All I have to do is just figure out what this delta x from b to c is. And whenever I get stuck with one of these variables, I have to go to a different interval in order to solve for it. So delta x from b to c, the only other thing I can relate that to is the total displacement. So that's gonna be that the the the total interval from a to c. So when it comes to this equation, the total interval, the total displacement, delta x from a to c, is really just the sum of the two parts From A to B and from B to C. I know that this distance over here, or this displacement, is 16 meters to the right from where I started. That's the total. So if I can figure out basically this, first displacement from A to B, which by the way is the red displacement, then that means I can figure out what this delta X from B to C is. Remember, I came over here looking for this variable, so I can plug it into this formula. So now all I need to do is if I'm trying to figure out what delta X AB is, I just go to that specific interval. So that's gonna be the interval from a to b. The equation is VAB equals delta XAB over TAB. And so delta XAB is gonna be VAB times TAB. So I have both of those numbers, it's 38. So this is gonna be 3 times 8, and that's 24. So now that I have this 24 over here, I just pop that into this equation. So 16 equals 24 plus something, and that's my unknown variable that I'm looking for. So if you just move this over to the other side, then what you're gonna get is that negative 8 actually, let's go ahead and write that in blue. Negative 8 is equal to delta x from b to c. And that makes sense. So this has to be negative 8 over here. So this makes sense because, remember, in this interval, we're walking backwards. So that displacement should be negative. So now all we do is we just pop this number into this equation over here and finish off that calculation. Or actually, sorry, this equation over here. So that means that delta of T from B to C is negative 8 over the velocity, which is negative 2. Again, don't forget that negative sign over there. And if you work this out, you're gonna get 4 seconds and it's gonna be positive because the two negative signs will cancel out, which makes sense. You should get a positive number over there. So that's our answer. We're walking backwards for 4 seconds. Let me know if you guys have any questions.

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You walk to the right at 3m/s for 8s, then turn around and walk backwards at 2m/s for some unknown time. You end up 16m to the right from where you started. How long did you walk backwards?

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