Average Velocity - Online Tutor, Practice Problems & Exam Prep

Velocity and speed are key concepts in physics, where speed is a scalar quantity defined as distance over time ($s=\frac{d}{\mathrm{\Delta t}}$), while velocity is a vector quantity defined as displacement over time ($v=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$). Understanding these differences is crucial for solving motion problems, especially when dealing with average velocity and multiple intervals of constant speed. Organizing variables systematically aids in problem-solving and enhances comprehension of motion dynamics.

Intro to Velocity and Speed

Hey, guys. In previous videos, we talked about the difference between distance and displacement to describe how far something moves. And in this video, we're going to talk about another pair of important, but related words called velocity and speed. Let's check it out.

So let's take a look at a familiar example. Let's say we were moving from A to B and then back to C again. We know that this is 10 meters to the right, and they're going to move 6 meters, and then we're going to be moving to the left here. There were two words to describe how far something moves. That was distance, which remember was given by the letter d, and was a scalar. Your distance is just the total of everything that you travel. So we know this totals 16 meters. The displacement, on the other hand, was a vector and it's the difference between where you started and where you ended, and it was the shortest path between those two points. And we know that your final position is 4, your initial position is 0. So your displacement, which is just 4 minus 0, was just 4 meters to the right.

So just like distance versus displacement, there are two words to describe how fast something is moving. And the basic idea here, guys, the main idea, is that speed is related to the distance that you travel whereas velocity is related to the displacement that's traveled.

Let's take a look at the equations. Speed is represented by the letter s and it's defined as the distance over time. Velocity is represented by the letter v and it's defined as the displacement over time. So the equations are as follows:

Speed = D Δt m/s

Velocity = Δx Δt m/s

That's going to be the units that we're working with over here.

Now, as speed is a scalar, it will always be either positive or 0 because it depends on the distance and distance can always just be positive or 0, it can never be negative. Whereas the displacement can be positive, negative, or 0, so your velocity can be positive, negative, or 0 because that's what your displacement can be. And so let's talk about negative velocity for a second. What does negative velocity mean? Remember, signs in physics just have to do with directions. So negative just means that you're moving in the opposite direction to whatever the positive direction is, and your problem is going to tell you what that direction is. So for example, let's say the right direction is positive, anything moving to the left is going to be negative. And sometimes, in rare occasions, you might see that the left direction is positive, your problem will tell you this, in which case anything that's moving to the right is going to be negative. So it just means whatever your problem tells you the positive direction is, anything moving to the opposite of that is going to be negative velocity.

Alright, guys. That's really all there is to it. So let's take a look at an example. We're going to jog 15 meters in 2 seconds, then 9 meters backwards in another 2 seconds. Let's calculate the speed and velocity. So, let's just draw a quick little diagram of what's going on here. So I'm going to move, I know this is going to be 15 meters in this direction. And then I'm going to move 9 meters backwards in another 2 seconds. So this is going to be 9 meters like this. So I know that the time here is going to be 2 seconds and the time here is also going to be another 2 seconds.

The speed is going to be calculated using the distance divided by ΔT. That's the equation that we have right up there. So, our total distance is 15 + 9 meters and our total time, is 2 + 2 seconds, making it 4 seconds total. So:

Speed = 24 4 ≈ 6 m/s

What about our velocity? Velocity is computed as Δx over ΔT for the whole trip. The shortest path from initial to final is actually from the start to where we ended after moving forwards then backwards. Thus, our displacement Δx = 15 - 9 meters, which equals 6 meters. Therefore:

Velocity = 6 4 ≈ 1.5 m/s

Since the displacement points to the right, the velocity vector also points to the right. That's it for this one, guys. This is the speed and velocity, and let me know if you guys have any questions.

Beginning from a signpost, you run 60m to the right, then 60m back. The entire trip takes 24 seconds. What is your speed and velocity for the whole trip?

• A. Speed = 5 m/s, velocity = 5 m/s
• B. Speed = 2.5 m/s, velocity = 0 m/s
• C. Speed = 5 m/s, velocity = 0 m/s
• D. Speed = 0 m/s, velocity = 0 m/s

Solving Constant and Average Velocity Problems

Hey, guys. So oftentimes in physics, you have to solve problems in which there is a constant or average velocity. In this video, I'll show you how to solve these problems by using a slightly different form of the velocity equation, which you're already pretty familiar with. Let's check it out.

So, guys, remember that velocity is measured between 2 points, an initial to a final position. So your textbooks might refer to this as an average velocity. Let's check it out. If you were going from 0 to 8, for example, in a span of 4 seconds, then your velocity is just delta x, which is a change of 8, over delta t, which is a change of 4. And so you get 2 meters per second. But this velocity is often called an average velocity because you don't really know what happens in between. All you know is that you went from 0 to 8, and it took you 4 seconds. But during the middle, you could have sped up to 3 meters per second, slowed down to 1, and then sped back up again to 3. You have no idea. If you were to plot this out on a velocity versus time diagram that shows your velocity on the y-axis and your time on the x-axis, it would look all squiggly like this because you're speeding up and slowing down. And this makes physics problems a nightmare to solve because you have to sort of account for all the speeding up and slowing down.

One way we can sort of go around this is we can say, well, we can kind of smooth out all of these velocities and pretend as if we were just traveling with a constant or average velocity that smooths out all of these different variations. So this is an average velocity. This makes things a lot easier in physics because average velocities behave like a constant velocity, one we already know. It's that v₊𝒯₎=ΔxΔt. That's it. That's the one equation that we're going to use.

Now we can take this equation, and we can actually manipulate it to solve for delta x because there are some problems where you have to work with delta x in the position. And so if we rearrange for this, we've got Δx=v₊𝒯₎⋅Δt. All I did was just move the Δt to the other side. And then some problems or some textbooks will start dropping this average over here. Now what we can also do is take this delta x and we know that this is just the x final minus x initial. So some textbooks and some equations will move this over to the other side, and what we get is we get this equation over here. X=X0+V⋅Δt. So again, this really is just a different form of this equation, kind of just expanded out. And it gets a special name because it gets used a lot in physics. It's called the position equation. So later on, when we're talking about the position equation, I'm talking about this guy over here. And we call the position equation because it tells us that your final position x is just your initial position X0 plus how far you've moved, that v times delta t. And so we're going to use this equation over here whenever we're solving for the velocity and we'll use this equation here whenever we're trying to solve for the positions.

That's all there is to it. Let's get some practice, guys. Alright. So we're going to solve for the unknown variables in each one of these diagrams here. We've got from going from 0 to 20 in 5 seconds and we want to calculate the velocity. So that means that we're just going to go ahead and use the velocity equation here. So that means that our velocity is going to be delta x over delta t. And so delta x is 20 minus 0. Right? That's my initial and final divided by a change in time of 5 seconds. So I just get 4 meters per second. That's pretty straightforward. Let's move on.

So now, I've got my initial position, velocity, and time, and I'm trying to find my final position. So now, I'm going to use my position equation just because it already gives me what the final position is going to be as a variable. So I've got X is going to equal X0 plus V times delta T. So my final position is my initial position of 2 plus the velocity of 3 and the change in time of 6 seconds. 3 times 6 is 18, 2 +18 equals 20 meters. So this is going to be my final position right over here.

And for the last one, let's check this out. Now we have to be a little bit careful because if you notice in the first two diagrams, we were moving to the right. But in this last one here, we're actually moving to the left. So we have to be careful with that because we know this is going to be the negative direction. This is going to be the positive direction, and that makes sense because our velocity ends up being negative here. But we're still just looking for the initial position now. So we're going to start off by using the position equation. X equals X0 plus V times delta T. So if I'm looking for this initial position, then I can just move all of this over to the other side. So I get X minus V times delta T is equal to X0. So let's just make sure I have everything. I know what my final position is. I know what the velocity is. What about the change in time Δt? Well, I have my final and initial times, that's not delta t. So I can just quickly go ahead and say, well, these 2 delta t's are going to be t final minus t initial. So this is just going to be 7 minus 3, which is 4 seconds. So now, I have my delta t, and now, I can go ahead and plug everything in. So my final position is going to be negative 20. You plug it in with the signs and be careful. Minus my velocity is negative 4. Again, be careful with the signs. And then delta t is 4 seconds. And so these end up canceling out, so you can make them both positive, and then you get negative 20 plus 16 is equal to or my initial position. And if you work this out, this ends up being negative 4 meters. And that's our answer. That's our initial position. Alright, guys. That's it for this one. That's 4 meters. Let me know if you guys have any questions.

Baseball Pitch

Hey, guys. Let's check this problem out. We have a baseball pitcher who can throw a baseball at some speed. We're trying to figure out how long it takes for the baseball to travel the distance to the home plate. Let's just draw a quick little sketch. I've got a baseball like this, and I know that the velocity that this is thrown at is 44 meters per second. Now, I'm not told that the baseball slows down, speeds up, or anything. So when this happens in problems, you can assume that this is a constant velocity. So it's a constant velocity that the baseball is traveling, and we know that the distance to the home plate, which is Δx, is going to be 18.5 meters. So now the only thing we have to figure out is what's the time. So what is the time? Well, we're working with constant velocity, which means we can only use one equation. Now that equation, remember, is that the whoops. It's the velocity is equal to Δx over Δt. And so what's the variable that I'm looking for? It's just ΔT. So I can just rearrange this really quickly. So I can move these basically to the other side. I can swap the v and the Δt. And what I get is the t is equal to Δx divided by the velocity. So that's just 18.5 divided by 44. And if you work that out in your calculators, you're going to get 0.42 seconds. So it's about half a second for the baseball to travel to the home plate. So for those of you baseball players out there, this actually might make some sense to you. Alright, guys. That's it for this one. Let me know if you have any questions.

t Δx v = 18.5 44 ≈ 0.42 s

Constant Velocity with Multiple Parts

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.

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?

• A. 8.0 s
• B. 2.3 s
• C. 16 s
• D. 4.0 s

Race Against Time

Hey, guys. Let's work out this problem together. We're told that a runner is 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. This is a classic motion with multiple parts, so let's just stick to 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 meters in 20 seconds. But there is a point where they realize they have to go faster, so there's a point right here where they're going to change speeds. Remember we have to label all of our intervals. This is going to be a, b, and c, and I'm going to list out all the variables for this one in red and this one in blue. So list out the variables. We know we're going to have Δx from A to B, we're going to have velocity from A to B, and T from A to B. Also, notice that in the problem, it says that we're in constant or average velocity for the whole entire time. So there's really only just those three variables to keep track of. Now for the second part, we have Δx from b to c. We have the velocity from whoops. That's the velocity from b to c, and then the time from b to c. And now finally, we remember we have the whole entire thing. There are 2 parts, but 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 meters. 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 Δx 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 going to be running for 4 meters per second for 14 seconds. So that means that the velocity from A to B is going to be 4 and the time is going to be 14, but we don't know what the distance is. In the second part, we actually don't know anything about this interval. We don't know 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 going to be our target variable over here. So now that we've drawn the diagram and we've listed out all the variables, now we just have to go ahead and solve by writing the equations for each of the intervals. Remember, we're working with constant velocity. There's only one equation in each interval that we can solve for. So the velocity from b to c is going to be Δxbtoc, so basically, all we have to do is just figure out, remember, ΔxBC is going to represent the distance left at the end of the for the end of the race, and the BC is the time that we need to finish it to in order to complete under 20 seconds. So I just have to go ahead and figure out what each one of these variables are in order to get that, that average velocity that I need. So whenever I'm stuck and I need to figure out what ΔxBC is, there's only one other equation that I can go and find. So if I don't have it over here, then I have to relate it to the total, the the total distance, which is ΔxAC. I know that this distance is 100 meters, but I also know that this distance is made up of both of these two distances added together. So basically, I can set up an equation. Δx from A to B plus Δx from B to C. And now I just have to figure out if I actually know 2 of those things. Obviously, I know the total distance is 100. But if we 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 ΔxAB 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 2 out of the 3 variables that I need, so I can actually solve for ΔxAB. So we start off the equation. V a to b is equal to ΔxAB over T a to b. And so I can rearrange for this really quickly, and I can get that ΔxAB is equal to VABTAB. And I actually know what both of these things are, so I can just go ahead and solve. This is going to be 4 times 14, which is going to be 56. So now I can just plug this back into this formula and figure out what that distance is. So I know that, Δ let me set up this a little bit differently. I have 56 plus ΔxBC is equal to 100. Right? The first part from the second part, plus the second part, is equal to 100. So that means that Δx from B to C is 100 minus 56, which is equal to 44 meters. So that's 44 meters, which I've got right there. So that's, that's the first variable that I need. And the second one that I need now is the time. So remember that there are 2 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 in the first part was 14 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 is needs to be is 6. Because I have to finish the race in 20 seconds. So if this is 14, then this has to be 6. So now I have everything I need. So a VBC is equal to 44 and then divided by 6. And if you work this out, you're going to get 7.3 meters per second. So they have to speed up to the speed in order to finish the race in that time. Alright, guys. That's it for this one. Let me know if you have any questions.

Here’s what students ask on this topic:

What is the difference between speed and velocity?

Speed and velocity are both measures of how fast an object is moving, but they differ in their definitions and properties. Speed is a scalar quantity, meaning it only has magnitude and is defined as the distance traveled over time. The formula for speed is:

$s=\frac{d}{\mathrm{\Delta t}}$

where d is distance and Δt is the time interval.

Velocity, on the other hand, is a vector quantity, meaning it has both magnitude and direction. It is defined as the displacement over time. The formula for velocity is:

$v=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$

where Δx is displacement and Δt is the time interval. This distinction is crucial in physics, as velocity can be positive, negative, or zero, depending on the direction of motion.

How do you calculate average velocity?

Average velocity is calculated by dividing the total displacement by the total time taken. The formula for average velocity is:

$v{\mathrm{avg}}_{}=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$

where Δx is the total displacement and Δt is the total time interval. For example, if an object moves from point A to point B and then back to point C, you would calculate the displacement from A to C and divide it by the total time taken for the trip. This method smooths out any variations in speed during the journey, providing a single value that represents the overall motion.

What does negative velocity indicate?

Negative velocity indicates that an object is moving in the direction opposite to the defined positive direction. In physics, the sign of velocity is determined by the chosen coordinate system. For example, if the positive direction is to the right, then a negative velocity means the object is moving to the left. The formula for velocity is:

$v=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$

where Δx can be positive or negative depending on the direction of displacement. Understanding negative velocity is crucial for solving problems involving motion in multiple directions.

How do you solve problems involving multiple intervals of constant velocity?

To solve problems involving multiple intervals of constant velocity, follow these steps:

1. Draw a diagram to visualize the motion.
2. List the variables for each interval: velocity (v), displacement (Δx), and time (Δt).
3. Use the formula $v=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$ to find unknowns for each interval.
4. Sum the displacements and times for all intervals to find the total displacement and total time.
5. Calculate the average velocity using $v{\mathrm{avg}}_{}=\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$.

Organizing variables systematically helps in solving these problems efficiently.

What is the position equation in physics?

The position equation in physics is used to determine the final position of an object given its initial position, velocity, and time. The equation is:

$x=x_0+v\mathrm{\Delta t}$

where x is the final position, x₀ is the initial position, v is the velocity, and Δt is the time interval. This equation is particularly useful for solving problems where you need to find the final position after a certain period of time, given a constant velocity.