Kinematics Equations - Online Tutor, Practice Problems & Exam Prep

Understanding motion problems involves mastering the four key equations of motion, applicable only under constant acceleration. These equations relate variables such as initial velocity (v₀), final velocity (v), acceleration (a), displacement (Δx), and time (t). Recognizing the signs of acceleration is crucial; positive acceleration indicates increasing velocity when aligned with initial velocity. Conversely, negative acceleration can still result in increasing speed if the initial velocity is also negative. Organizing variables and selecting the appropriate equation is essential for solving complex motion scenarios effectively.

Kinematics Equations

Hey, guys. You'll need to know how to solve motion problems involving acceleration. So what I'm going to do in this video is introduce you to the 4 equations of motion or sometimes called the kinematics equations that you absolutely need to know. But more importantly, I'm going to show you when and how to use each one of these equations. So this is really important for you to learn, practice, and master because it sets the stage for the rest of physics. So let's check it out.

So guys, remember for constant velocity, when the acceleration is equal to 0, for motion problems, the only equation we could use is v=ΔxΔt. We had another version of this equation that we called the position equation. Really these two things were just 2 versions of the same equation. Well, a lot of times in physics, you'll find that the acceleration is not zero. And in those cases, you're going to need 4 more equations called the equations of motion, sometimes called kinematics equations or uniformly accelerated motion equations. Now uniformly accelerated motion just means that the acceleration must be constant. And that's what you absolutely need to know about these equations. You can only use them when the acceleration is constant. Now a lot of textbooks will show the derivations and the proofs and all that stuff, which you don't really need to know, but you should memorize them. So I have them listed in this table here. So we're going to talk about each one of these equations. So let's start with the first one, which is that v = v₀ + at. This equation actually comes from an equation that we're pretty familiar with. It comes from a, the acceleration average, is equal to Δv Δt . So you can manipulate this equation to look like this in the same way we can manipulate the v=ΔxΔt to look like the position equation. It's the same idea.

So we have the next one here, which is nothing really, nothing special about it. v2 = v₀2 + 2aΔx. This third equation here I've written in 2 different ways because you'll commonly see it written in those two different ways. They look the exact same. The only difference is that this Δx over here can sometimes remember, it can be expanded and rewritten as x minus x_initial, and they just move this thing to the other side. So again, it means the same exact thing. You'll often see them, sort of written interchangeably like that. So now the last one over here is Δx = v₀ + v2t. And this one has a little asterisk because in order to use this, I strongly recommend that you look at your you ask your professors because some textbooks and professors might not allow you to use this. But basically what this equation says is this, v₀ + v is actually just the average of 2 velocities. So what this is really saying here is Δx = v_avg⋅t, which we actually already know.

Okay. So that's really it. So what the important thing you need to know about these equations is that all of them have some combination of all of these 5 variables. And you need to know which equations have what variables in order to get the right answer. So let's go through each one of them really quickly. So we have v = v₀ + at. So we have v, v₀, a and t, but Δx is missing. So I'm going to write the little sad face there. Now we have v, v₀, a, and Δx. So that means this has v, v₀, a and Δx. But it's missing time. And now for these two equations, for number 3, whatever form you're using, this has Δx, it has v₀, t, and a. So this has, Δx, v₀, t, and a, but it's missing the final velocity. And now finally, we've got Δx = v₀ + v2t. So that means it has Δxv₀vt, but it's missing the acceleration. So notice that there's a pattern here. Every single one of these equations is missing one of the variables, except the only one that I have it all in common is the initial velocity. So guys, what you need to know about these 5 variables here, is that to solve any motion problem, whether it's a car or a rocket or whatever it is, with these equations, you're always going to need 3 out of the 5 variables to solve. So the whole game here with all these problems is figuring out which of these 5 variables you have and which 3 you can figure out in order to pick the right equation to get whatever you're missing. So in order to show you how that works, let's just go through this example together.

We have a racing car that is starting from rest. It accelerates constantly, which means that we're able to use the UAM equations, which is good. We have a 160 meter track and the car crosses after 8 seconds. We're going to figure out what's the acceleration of the car. So all of these problems, we can always follow these list of steps to get the right answer. Basically, we're going to start off by drawing the diagram and listing off our variables. It's a great visual way to figure out what we have and what we need. So we've got the, initial velocity. You're saying it starts from rest. We got this car here. It's going to travel and it's going to cross the finish line and then it's going to be moving, probably with some final velocity over here. So let's list off our variables. So let's start off with Δx, then we've got v₀, we've got v, a and t. Now, basically, now that we've drawn and listed the 5 variables, let's just identify what we know and what our target is. We're told that the track is a 160 meters, so that's the length, that's 160. The time is going to be crosses the finish line after 8 seconds, so that's a time, that's t equals 8. And so what are we trying to find? We're trying to find the acceleration. So this is going to be our target variable a. So that's going to be the question mark. So that's what we've got here. But notice how we only end up with 2 out of the 5 variables. Now whenever this happens, there's going to be a clue inside of the problem that's going to give you that third one. So let's look out here. We have a racing car that starts from rest and accelerates constantly. So what you need to know there is from rest means that the initial velocity is equal to 0. It starts off with a v₀ of 0. So that's that third variable that you need. So our v₀ is equal to 0 and now we have 3 out of 5 variables. So we're good to go. We can pick an equation. So now the next step here is we have to pick an equation without this ignored variable. So let's take a look at what happened here. I was given 3 variables, whether numbers or words in the problem. I'm asked for one of them. But this final velocity, I have no information about. It's not asked and it's not given. So this is called the ignored variable over here. It's a variable that is not asked for or given. And so whenever you are picking the equation, you have to pick the equation that contains the variable you're looking for. So for instance, I'm going to pick an equation that contains acceleration, but it excludes the ignored variable here. So basically, I'm going to pick an equation that does not have v inside of it. So let's take a look at our list and figure out which equation that is. Well, the first one has v, so it's not going to be that one. The second one says v2, so it's not going to be that one. The third one, has a, but it does not include v, which is good. And just to be, you know, thorough, the 4th equation has v+v₀ over 2, so that's also bad. So notice how this always will give you just one equation to use. So from this list here, we're going to pick equation number 3. Δx = V₀ T, + 12 AT2. So now, let's just go ahead and fill out all the variables. So all the numbers. We've got 160. Initial velocity is 0 times 8, so that just goes away. Plus 12, now we have a, now we have 8². Notice how we're only ending up with one unknown variable. That's the acceleration. That's what we want. Then we just rearrange and solve. So this 12 goes to the other side, the 8² goes to the other side. So I have a 160 times 2 divided by 8² equals the acceleration. If you go ahead and plug that in, what you're going to get is 5 meters per second squared. So that's our answer over here.

Moving on to part b. So actually that was part a. So part b now, we're going to be looking for what the car's velocity is at the finish line. So in other words, we're going to be looking for what the final velocity is. This is actually the ignored variable in part a. So we're looking for the final velocity over here. You can go through the list of steps. We already have the diagram. We have the 5 variables. We actually know what this acceleration is now. Now it's just equal to 5. Target variable is v. So actually there is no ignored variable anymore because we have 4 out of the 5. That's what makes these problems easier, is that as you continue to solve, you figure out more and more of the variables. So really we can start off with any one of the equations and the easiest one is going to be the first one. So, I'm looking for the final velocity, so I need v₀ + a times t. I know v₀. That's just equal to 0. I have the acceleration from the first part and I have time. So I'm just going to plug this stuff in. Really straightforward, I've got v equals 0 plus, and then I've got 5 for my acceleration, and then 8 seconds. So we end up with a final velocity of 40 meters per second.

Alright, guys. So that's it. We're going to get a lot more practice with this. That's it for this one. Let me know if you have any questions.

A car accelerates from 5 m/s to 21 m/s at a constant 3.0 m/s². How far does it travel while accelerating?

• A. 437 m
• B. 2.67 m
• C. 69.3 m
• D. 42.7 m

A moving hockey puck encounters a patch of rough ice and slides 90cm before coming to a full stop. Assuming a constant deceleration of 8 m/s², what was the puck's initial speed?

• A. 3.79 m/s
• B. 2.68 m/s
• C. 37.9 m/s
• D. 1.90 m/s

Certain rifles can fire a bullet with a speed of 970 m/s just as it leaves the muzzle (this speed is called the muzzle velocity). The muzzle is 70.0 cm long and the bullet is accelerated uniformly from rest within it. 

For how long (in ms) is the bullet in the muzzle?

• A. 144 ms
• B. 1.44 ms
• C. 0.722 ms
• D. 1440 ms

Acceleration Signs

Hey, guys. So in your motion problems, you're going to run across this phrase, "speeding up" or "going faster," and you might take that to mean that your acceleration is positive, but be careful because this does not always mean this. Students get tripped up all the time. So in this video, I'm going to show you how the acceleration sign affects velocity and speed. I'm going to show you a really simple way to think about this. Let's check it out. So guys, again, positive acceleration does not mean speeding up or going faster. Positive acceleration literally just means that your velocity is becoming more positive. And on the other hand, negative acceleration just means that your velocity is becoming more negative. And we can just see this from the equation. We know that acceleration is ∇v∇t. So positive acceleration means that the change in velocity is positive, aka your velocity is becoming more positive. And then negative acceleration means that your velocity is becoming more negative. That just comes from the equation. Let me just show you a bunch of examples to see how this works and where the confusion comes from so you don't make the same mistakes yourself.

So we have all these problems here. We're going to indicate whether the acceleration is positive or negative and if the speed is increasing or decreasing. We're going to assume ∇t = 4. So let's get to it. We've got vv0=10, vfinal=30. So I'm just going to draw a quick little sketch. Here's my v0 which is 10. Then at some later time from here to here, I know that my final velocity is going to be a bigger vector, and this is going to be 30. So to calculate the acceleration, I need ∇v∇t. So my change in velocity is 30 minus 10, that's my initial divided by ∇t which is 4. So this is 20 over 4, and I get 5 meters per second squared. So notice how this is a positive number, that's my acceleration, so that's positive. What about the speed? Well, whenever you think about the speed, just think of the number of the velocity, the magnitude of velocity, not the actual sign. So I'm going from 10, that's this number, to 30, which is a bigger number. Again, forget about the sign, which means that my speed is increasing.

Let's take a look at part b. Now I have a velocity that's negative and a final velocity that's negative. So I'm actually going to start going to the left. So this is my initial velocity here. My v0 is negative 10. And then my final velocity over here is going to be negative 30. And I know that this takes some time, which is ∇t. So to calculate my acceleration, I need ∇v∇t. My ∇v is going to be negative 30, that's my final minus my initial negative 10. Be careful with the signs. And then ∇t is 4. So this is actually negative twenty over 4. So this is actually negative 5 meters per second squared. So my acceleration's negative. So this is my acceleration over here. What about the speed? Well, again, just think about the number, not the sign. Just think about the magnitude of your velocity. I'm going from negative 10, so the number is 10 to negative 30. The number is 30. The number gets bigger. Forget about the sign. So that means my speed is increasing. So notice how in one situation, my acceleration was positive, my speed was increasing. And in the other situation, my acceleration was negative, but I was still speeding up, and my speed was still increasing. So speeding up just means that you're going faster. And that means that the magnitude of your velocity, just think about the number, is increasing. And so the easiest way to think about this is if your acceleration and your initial velocity are of the same sign, that's when your speed increases. For example, here, my acceleration was positive. My initial velocity was positive, so my speed increased. Over here, my acceleration was negative. My initial velocity was negative, and so my speed increased as well.

Let's take a look at more problems. So here we've gotten a velocity of 30 and a final velocity of 10. So here, now I've got my initial velocity is 30, and then my final velocity is going to be 10 over here. So I'm going to draw up my little interval from here to here. Now, I just calculate the acceleration. A=∇v∇t. My ∇v now, my final velocity is 10, my initial velocity is 30, and my time is 4. So this is going to be negative 20, and this is going to be 4. So I get an acceleration of negative 5 meters per second squared. So my acceleration is negative. What about the speed? Again, think about the number. First, I'm going from 30, and now I'm going to 10. The number is getting smaller this time. So my speed is actually decreasing. Let's go to the last one, negative 30 to negative 10. So here what happens is, first I'm going, this is my v0, it's negative 30. Draw it again. Negative 30 over here. And then finally, it's going to be negative 10. And then my interval is over here. I know that's 4 seconds. So what's my acceleration? So acceleration, ∇v∇t. So my ∇v, what's my final velocity? It's negative 10. My initial velocity, negative 30. Again, keep track of all the signs. Over 4. So this becomes positive, and positive. This becomes 20 over 4, and that's just 5 meters per second squared. And it's positive. So here we got a positive acceleration. What about the speed? The speed is going from 30 to 10, the number, just forget about the sign, the number is going from 30 to 10, which means that my speed is actually decreasing here. So again, now we have a situation where we have negative acceleration and decreasing speed, but positive acceleration could also mean decreasing speed. So slowing down just means that obviously, you're going slower, which means that the magnitude of your velocity is decreasing, and this happens whenever your acceleration and your initial velocity have the opposite signs. So for example, here, my acceleration was negative, positive initial velocity, so I slowed down. Here, my acceleration was positive, initial velocity was negative, and so that means I am also slowing down. Alright, guys. Hopefully, this illustrates where that confusion comes from and so hopefully, these rules here give you a better understanding of what the relationship between signs of acceleration and velocity are.

Alright. I've got this one last problem here. Let's just get to it. We've got the driver of a truck that is moving to the left at 30 meters per second, slows down by taking their foot off the pedal. And basically, we're going to calculate the magnitude and direction of the acceleration, and it's assumed constant. So we're actually just going to use our normal steps for solving motion problems with acceleration. So let's just get to it. Let's just draw a quick little sketch here. So I've got this truck that's moving to the left. So here, I'm at this truck that's moving to the left. We know that this initial velocity here is going to be 30 meters per second, but if it's moving to the left, it's actually going to be negative. And so, we're told that the truck comes to a stop after traveling 150 meters. So basically, after some distance over here, which we know ∇x is going to be negative 150. Now, at this point here, the final velocity is going to be 0. And again, this is negative because we're moving to the left. So we need 3 out of 5 variables. Let's just go ahead and list them out. I need ∇x, I need vnaught, v, a and t. I know ∇x, is negative 150. I know my initial velocity is negative 30, and my final velocity is 0, and then my a is what I'm looking for. This is my target variable and then, yeah. So I've got my diagram, and I've got my 5 variables. I've got the known variables and what my target variables are. So now, I just have to pick the UAM equation that does not have my ignored one. So these are my 3 variables that I know. I'm looking for the acceleration, and the time is going to be my ignored variable. So if you look through our list of equations here, the one that doesn't include time is actually going to be the second one over here. So we're going to use equation number 2. Whoops. So vfinal2=vinitial2+2a∇x. And so we're looking for this acceleration over here. We know that my final velocity is going to be 0, 02. My initial velocity is going to be negative 30, but it's squared, and then plus 2 times a times ∇x. So I get 0 equals this, being 900 + 2 times a times my ∇x, which is negative 150. So this ends up being, when I move this to the other side, I get negative 900 equals, and then when I multiply 2 and the negative 150, this becomes negative 300 times a. And so, all I have to do now is divide, and basically, my a is going to be 900 over 300, which is just 3 meters per second squared. What about the sign? Well, now my sign is positive. It's a positive 3 meters per second squared, but notice how everything in my diagram was negative. So even though I was slowing down by taking my foot off the pedal, my acceleration was actually positive, and that's because everything in my diagram was pointing to the left. Alright, guys. That's it for this one. Hopefully, this sort of clears up some of the confusion. Let me know if you have any questions.

Acceleration with Multiple Parts

Hey, guys. So for this video, we're going to check out how we solve problems where an object is moving under constant acceleration in one or multiple parts. And what we're going to find is that it's exactly like how we solve any motion problem in general. So let's check it out. Now, guys, remember in these problems, organization is our friend here. So, the situation's going to be different. You got a guy running and speeding up or slowing down. But when there's multiple parts, the first thing you want to do is just draw the diagram and then list the 5 variables like any motion problem. So we have parts a to b and then b to c, and then we also have the whole entire interval over here. And now we just have to draw, and now we have to list our 5 variables. So we've got Δx, initial velocity, final velocity, acceleration, and time. But because there's going to be sets of them, I'm going to have to assign them variables. So I'm just going to give them little a to b. Now my initial velocity is actually going to be the velocity right here, so I'm going to call this v_a. Final velocity is going to be right here, so I'm going to call this v_b. And then we've got the acceleration and the time. Now, we just do the exact same thing for the second set. So Δxb to c. Now, my initial velocity is v_b, my final velocity is v_c, and then the acceleration and time. And now finally, you also remember we have the 2 different parts, but we also have the whole, the whole entire interval. And the only two variables that matter there are Δxa to c, and then Δta to c. So it's the total distance and the total time. Usually those are the two ones that are given, or that are asked for. So once we have this sort of organizational structure for all of our problems and we have the list of variables, now the rest of it is just picking equations. So let's just work out this example together and see how this stuff works. So we're told that we're driving at a constant 30 meters per second, and then we see a road hazard and it takes 0.7 seconds for us to react and stop. But then we decelerate and then come to a stop. So that means that there are 2 different intervals going on here. So the first step is always to set up the problem by drawing the diagram and then listing your 5 variables. So let's go ahead and do that. So I've got this little diagram here. This is the part where I haven't yet slammed the brakes, and it takes 0.7 seconds for me to react. So this is the reacting part. And then also, over here, I've got once I've slammed the brakes, now I'm coming to a stop. So this is the stopping part. So now we're just going to go ahead and list the 5 variables, Δxfrom a to b, once we've actually labeled them. Right? This is a, b, and c. And now I've got my v_a, v_b, acceleration, and time. Okay. So I don't know the distance for this first part. All I know is that I'm driving at a constant 30 meters per second, which is my initial velocity. So this is right here. And because it is a constant 30 meters per second, that's also going to be my final velocity when I get to the end of this first interval. Which means that the acceleration is equal to 0. So acceleration from a to b is 0. It's the same exact velocity. And we also know that it takes 0.7 seconds for us to react and stop. So those are our numbers there. Now let's do the second, the same thing for the second interval. I don't know the distance from b to c. I don't know the stopping distance. All I know is that the initial velocity here is going to be 30 meters per second because that's what it is over here. And then I know that I'm going to come to a stop. So a stop means that the final velocity is going to be 0. And I know that the acceleration in this point is going to be 6 meters per second, but it's decelerating. So it's going to be negative 6. And then finally, the time, I don't know how long it takes for us to come to a stop. So these are all of our, basically, our variables here. So the second step is for each part, we just have to identify the known and now with the target variables. So what is part a asking us for? It's asking us for how far we're traveling before we're applying the brakes. So which one of these 10 variables is that going to be? Well before the breaks is actually in this first part over here, and it's asking us for a distance. So it's actually asking us for Δx_{A to B}. So that's that second step there. So we know what we're looking for, Δx_{A to B}. Now the third step is just to pick the right equation that does not have our ignored variable. But it's a little tricky here. It's a little weird here because we actually have 4 of the 5 variables and more importantly, the acceleration is equal to 0. And so if we go to our list of equations here, remember that we have 2 kinds of motion equations. When the acceleration is equal to 0, there's only just 1. It's constant velocity. So we actually don't have to pick any equations here, because there's really only just one to use. So that's V_AB = Δxfrom a to b, over T_{a to b}. And if we can actually rearrange for this,Δx_{a to b}= V_AB * T_{a to b} . So this is just going to be 30 times 0.7. And if you work this out, this is going to be 21 meters. So that's the answer. So it's just 21 meters and that's the distance. So let's move on to part b now and go through the list of steps. We've already drawn

Subway Train

Hey, guys. Let's check out this problem together. We've got a subway train and it's going to start from rest and accelerates, then it's going to travel at constant speed for some time, and then finally, it's going to slow down until it stops at the next station. So there's a bunch of moving parts here with multiple accelerations. So we're going to solve this like any other problem with multiple parts with accelerations. The first thing we have to do 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, I'll call it A to B. Then it's going to travel at constant speed. So from B to C, it's going to be constant speed. And then finally, it's going to stop as it approaches the next station. So this is actually going to be from C to D and it's going to 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 Δx from A to B, which we know, which we actually don't know. We don't know Δx from A to B. That's the distance in the first part. We've got the velocity starting from rest, so this is just going to be 0. And let's see, we, so we know this is 0. We don't know the final velocity, but we do know the acceleration and 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 going to 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 going to accelerate, which means it's going to speed up in the right direction, so that's going to be positive. So it's going to be 1.5 positive. Okay. For the second part, now we've got constant speed for 60 seconds. So we've got Δx from B to C, we don't know what that is. Vb, we don't know what that is either because we don't know what it is from the first part. VC, don't know what that is. Acceleration from B to C is going to be, well, it says constant speed. So this constant speed here actually means the acceleration is equal to 0 for this part. But we do also know that the time for part B is 60 seconds. So whatever this velocity ends up being, VB and VC, they're actually going to be the same thing because there's no acceleration. Alright. And so finally, now we have the stopping part. So we have Δx from C to D. We don't know what that is. So we have Δx from C to D, we don't know what that is. VC, we don't know what that is. Vd, that's just going to be when it stops at the next station right here. So this is going to be 0. And then finally, the acceleration from C to D, well, it's going to slow down at 3.5 meters per second squared. So first, it accelerated then it went to constant speed and then it's going to slow down which means that this acceleration is going to 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 out what's the total distance that the train covers. So remember, we have all of the different parts of this motion here, but we also have the whole entire interval. And this whole entire interval here, there's a distance, Δx 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 Δx from A to D is our target variable. How do we figure that out? Well, we don't have an equation for Δx from A to D using our motion equations, but we can figure it out by piecing together all of the different distances in each of the parts. So, right, if this was 10, 20, and 30, you just add them all together and that would be your total distance. So this guy is actually going to be Δx from A to B, plus Δx from B to C, plus Δx from C to D. So we're going to 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 going to look for Δx from A to B. So this is going to 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 UAM equation, one of our motion equations that doesn't have the ignored variable. Notice how I have 3 out of 5 variables, 1, 2, 3, and which means the one I'm ignoring is actually going to 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 going to be Well, let's see. We have VB here. This is final velocity, final velocity and this is also going to be final velocity. So we're actually going to pick equation number 3. So this is going to say that my Δx from A to B, which is actually my target variable, which is great because this is only going to be on the left side of the equation, is going to be my initial velocity times time, but we know this is going to be 0 because our VA is equal to 0, plus one half acceleration which is 1.5 times T squared. So this is going to be my 14 seconds squared. So I can just actually plug that straight into my calculator and my Δx from A to B is going to be 147 meters. So that's one of my variables, 147. 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 going to be my target variable here, Δx from B to C. So we'll go and focus on that. So how do I figure this out? Well, my Δx from B to C. So which equations am I going to use? Well, remember that we only use the 4 UAM equations whenever the acceleration is constant and it's not 0. When the acceleration is equal to 0, then all of our equations simplify and we only just use 1, which is V equals Δx over Δt. So if we're trying to figure out Δx from B to C, we're just going to use the constant velocity formulas. My VB to VC is equal to Δx from B to C over TB to C, which means that Δx from B to C is just going to 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. All I have to do is just figure out what the velocity is from B to C. And notice how it's actually going to be the same velocity. It's going to be either VB or VC because it's going to be a constant velocity. Right? So whatever I figure out for VB is going to be the same thing for VC. 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 going to have to go to a different interval and figure that out. Which other interval also has VB? Well, these either well, actually, there's only one choice. It's going to be the first part right here. So what we could do is we can actually use the first interval, because now we know 4 out of the 5 variables, to figure out what my velocity is that I ignored in the first part. So that's what we're going to go ahead and do. So now what I can do is I can go back over here and I can say, well, now I want to figure out what my velocity B is. So how do I do that? Well, now I'm going to use my 3 out of 5 variables for this part here. And if I'm trying to figure out the final velocity, I can just use the first equation. That's the easiest one. So I'm going to use equation number 1 to figure out the final velocity. So VB is equal to VA plus A times T. So I know I'm going from rest, so this is 0. And then my acceleration is 1.5 and my time is 14, so I get 21 meters per second. So that means that I know that this is 21 and this is also 21, which means now I can go ahead and use my formulas. Now I have the velocity and I have the time which means I can figure out Δx. So this is just going to be 21 times my TBC which is 60. And if you work this out, you're going to get 1260. So notice how sometimes you're going to have to stop and you're going to 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. Think about it like a piece of a puzzle. I've got the second piece of the puzzle, so 1260. So now I'm just going to have to figure out the last one. So the last one is Δx from C to D. So I've got to figure out this guy over here. Notice how also I figured out another variable when I did this. So remember how VC was an unknown? Well, we just figured out that VC was 21. Here it traveled at 21 meters per second, so that means it ended right here and now it's 21 meters per second over here and then eventually it's going to stop. So now we're looking for the Δx from C to D. We just go ahead and look at our UAM equations. We know there is an acceleration here, so we are going to use these variables, which means we do need 3 out of 5. I've got 3 out of 5, 1, 2, and negative 3.5, which means that my T is going to be my ignored variable over here. And so now which equation we're going to use? Well, the one that T is ignored in is going to be equation number 2. So this is the one we're using for 2, for Δx from C to D. So this is just saying that my final velocity VD squared is equal to my initial velocity VC squared plus 2 times A from C to D times Δx from C to D. So this is actually what I'm looking for here, my Δx. So now you just work everything else out, you plug everything else in. So we know my VD is going to be 0, my VC is going to be 21, and then 2 times the acceleration. Well, the acceleration is going to be negative 3.5, remember the negative sign, and then times Δx from C to D. 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 going to get negative 441 equals and then you multiply these things out, you're going to get negative 7 times Δx from CD. So now, my Δx from C to D is just Well, the negatives will cancel and you're just going to get 441 over 7 which is equal to 63 meters. And so this is the final piece of the puzzle, the final displacement. So now you just go ahead here and put them all back together again. So 63. I can say that Δx from A to D, the total distance traveled of all of the parts is going to be 147 + 1260 + 63. You add all these things together and what you get is you get 1470 meters 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 going to get totally lost when you're trying to figure these out. Anyway, let me know if you have any questions. That's it for now.

Here’s what students ask on this topic:

What are the four kinematic equations and when should each be used?

The four kinematic equations are used to solve motion problems under constant acceleration. They are:

1. $v_f=v_0+at$ - Use when you need to find final velocity.

2. $v_{f}2=v_{0}2+\; 2a\mathrm{\Delta x}$ - Use when time is not given.

3. $\mathrm{\Delta x}=v_{0}t+\; 0.5at2$ - Use when you need to find displacement.

4. $\mathrm{\Delta x}=\; 0.5(v_0+v_f)t$ - Use when acceleration is not given.

Choose the equation based on the known and unknown variables in your problem.

How do you determine the sign of acceleration in kinematics problems?

The sign of acceleration depends on the direction of the velocity change. Positive acceleration means the velocity is becoming more positive, while negative acceleration means the velocity is becoming more negative. Use the formula $a=\mathrm{\Delta v}\mathrm{\Delta t}$ to determine the sign. If both initial velocity and acceleration have the same sign, the object is speeding up. If they have opposite signs, the object is slowing down.

How do you solve a kinematics problem involving multiple parts of motion?

To solve a kinematics problem with multiple parts, follow these steps:

1. Draw a diagram and label each part of the motion.

2. List the five variables (Δx, v₀, v_f, a, t) for each part.

3. Identify known and unknown variables for each part.

4. Use the appropriate kinematic equation for each part, ensuring you account for the transition between parts.

5. Solve for the unknowns step-by-step, using the results from one part as inputs for the next.

What is the difference between velocity and speed in kinematics?

Velocity is a vector quantity that includes both magnitude and direction, while speed is a scalar quantity that only includes magnitude. For example, a velocity of 30 m/s to the right is different from a speed of 30 m/s, which does not specify direction. In kinematics, velocity is used in equations to account for direction, whereas speed is used to describe how fast an object is moving regardless of direction.

How do you find the final velocity of an object given initial velocity, acceleration, and time?

To find the final velocity (v_f) of an object given its initial velocity (v₀), acceleration (a), and time (t), use the kinematic equation:

$v_f=v_0+at$

Plug in the values for v₀, a, and t to solve for v_f. This equation is useful when you need to determine how the velocity of an object changes over time under constant acceleration.