Hey, guys. So a common type of problem that you'll see in motion involves an object that is moving with different but constant velocities in multiple parts. For example, let's say we had a guy that was walking from a to b with a constant velocity then picks up speed with a different but constant velocity from b to c. What makes these problems hard is not the math because at the end of the day, there's no acceleration in these problems. So we're only going to be using this equation here and three variables. What makes these problems hard is actually organizing which variables you have and which ones you need. So I'm going to show you a list of steps down here that we're going to use to solve any one of these problems. Let's check it out. And the key to solving these problems is again organization. So we're going to list off the three variables after we draw the diagram. We're going to do this every single time, and we've got three variables to keep track of: V, Δx, and Δt. But because there are two sets of them, what I'm going to do is I'm going to attach letters to every single one of them. So this is going to be v from a to b, Δx from a to b, and t from a to b. Do this every single time. V from b to c, Δx from b to c, and t from b to c. Now, you'll never lose track of which variables you have and which ones you need. There's also, before we move on, actually a third interval that we have to keep track of. There's not just the two pieces. There's also just the whole from start to finish. And there's also equations over here and variables that we need. V from a to c, Δx from a to c, and Δt from a to c. So this is what we're going to do for all of our problems. Draw the diagram, list out the variables, and then figure out what we need to know. Let's go ahead and solve the problem.
So we've got a car that's traveling at a constant 50 meters per second for 10 seconds, and then it changes and it goes at 30 meters per second for 600 meters. So this is constant velocity, multiple parts. So we're going to list off the intervals a, b, and c, and we're going to draw the diagram and then list out the variables for all the pieces. So we got v from a to b, we've got Δx from a to b and we've got Δt from a to b. I know the velocity in this first part here is 50 meters per second, and I know the time is equal to 10 seconds, but Δx, I don't know. Now from b to c, do the same thing. V from b to c, I know is 30 meters per second. Δx from b to c, I know is 600 meters here, but the time is something I don't know. Δt from b to c is unknown. So let's take a look at this first part here. The first part asks us to find the total distance traveled. Which variable is that? Well, remember there's a third interval here. It's the one from start to finish. We need the three variables here. V from a to c, Δx from a to c, and Δt from a to c. So this is actually the variable that we're looking for. What's the total distance of all the parts combined? And so this is going to be our target variable here.
Now let's move on to the second step, which is where write equations for each one of the intervals here. Now there is only just the one constant velocity equation, but there are a couple of more equations that we can write. For instance, this total distance traveled here, if you have multiple parts, is really just the distance that you cover in the first half from a to b, plus the distance that you cover in the second half from b to c. For an example, if this was 10 and this was 20, then the total distance was 30. That's just an example. So let's see here. My total distance from a to b is actually unknown. So I've got a big question mark here, but then my total distance from b to c is equal to 600. So if I can figure out that first half, then I can figure out the total over here, and that's what we're going to go ahead and find out. So I've got to figure out what this Δx from a to b is. So now I'm just going to go ahead and write the equations for the interval from a to b. There's only one equation that we're going to use. So we got these three variables here. My equation is: V AB = ΔxAB ΔtAB That's just the average velocity formula. Now what are we looking for? Remember that we came to this interval because we're looking for Δx from a to b to plug it into this equation. So we just have to solve for that. Δx from a to b is equal to V_AB * Δt_AB. So in other words, my v is 50 meters per second and my time is 10 seconds. So this is just a displacement of 500 meters. So now, I can plug this back into this formula over here. I get 500 and 500 plus 600 gives me the total distance traveled of 1100 meters. So this is my answer over here. So that's my answer of 1100.
So let's move on now to the second part, the average velocity from start to finish. Which variable is that? Well, the average from start to finish is again going to be in the interval from a to c. So just now this was from a to c. This is also from a to c. So this variable here that I'm looking for, this was the total distance traveled that was Δx from a to c. This is my average velocity from start to finish. That's going to be this variable over here. So now I'm just going to go ahead and write the equations for each one of my intervals. V AC = ΔxAC ΔtAC And so, if you take a look at this equation here, I just figured this part out actually in the first half. This is just the 1100 that I just solved for. Now I just need to figure out my Δt_AC which I don't know. So now, I just need a new equation for this Δt_AC. Well, if the total distance traveled was basically just the sum of the distance over here and over here, the total time that it takes for both of these intervals is going to be the same thing. What's the total time from a to b, plus what's the total time from b to c? So if this was 10 seconds and 20 seconds, then the total just, you know, the total time taken was 30 as an example. So now, let's look here because now I actually have t from a to b. I know that this is just 10 seconds over here. So I've got 10 plus now, I just need the time from b to c which I don't know. So now, I just need to go and get a new variable in order to get my total time and then I can plug it back into this equation over here and solve for the velocity. So I know that this is going to be 1100 divided by whatever I get for that variable over here, and this is going to be my final answer. Alright? So, basically, I need to go ahead and get this t from b to c over here. So I'm going to write the equation for this interval. ΔxBC = V BC · ΔtBC So I know now this is 20 seconds, which means I can plug it into this formula over here. This is 20 seconds. And so now I can add these two things together. So now, the total time that it takes for the whole trip is just 30 seconds. And so now that I have both of these numbers here, now I can get my final or average_velocity for the whole entire piece. It's 1100 divided by 30, and this is going to be 36.67 meters per second. So this is my final answer. Alright, guys. That's it for this one. Let me know if you have any questions. Let's get some more practice.