Understanding Kinematics: Motion Principles

In this video, we're going to talk about kinematics, which basically describes how objects move without any references to force. Now we're going to focus on kinematics in one dimension, mostly along the x-axis, but we can also work on some problems along the y-axis. When you get into two-dimensional kinematics, it covers projectile motion, which is a topic for another day. Now the first thing we need to talk about is the difference between a scalar quantity and a vector quantity. A scalar quantity is something that has magnitude only, whereas a vector quantity has both magnitude and direction. So for instance, mass is a scalar quantity. You could say that a book has two kilograms of mass. The two kilograms would be the magnitude of the mass of the book. But you wouldn't say the book has two kilograms of mass east. Direction wouldn't apply for mass. So mass cannot be a vector quantity. Distance is a scalar quantity. And displacement is a vector quantity. Displacement, you can think of it as distance with direction. So let's say if a person traveled 15 meters, you're basically describing a person's distance because you didn't mention direction. But let's say if someone traveled 15 meters east, now you're describing displacement. You're describing not only how far he traveled, but also where he traveled. So think of displacement as distance with direction. Now technically, displacement is really the change in the position of an object. So along the x-axis, you could describe displacement as the final position minus the initial position. Speed is a scalar quantity. Speed describes how fast an object is moving. Velocity is a vector quantity. Velocity is speed with direction. So if you were to say a person is traveling at 20 meters per second, you're describing the speed of the person. This is the magnitude. But let's say if a person is moving at 30 meters per second north, you're describing velocity because You've mentioned his speed along with the direction. So you have both magnitude and direction, which makes it a vector quantity. Now what about temperature? Would you say that temperature is a scalar quantity, or would you describe it as a vector quantity? So think of units of temperature. Let's say it's 80 degrees Fahrenheit outside. or 25 degrees Celsius. You wouldn't say it's 80 degrees Fahrenheit north. Direction doesn't apply for temperature. So because you can't describe the direction of temperature, it's not relevant. Temperature is a scale of quantity. You can only mention its magnitude, such as how hot it is, 80 degrees Fahrenheit, but you can't attach direction to temperature, which makes it scalar. Now acceleration, that is a vector quantity. Acceleration tells you how fast the velocity is changing. And you could say a car is accelerating at 2 meters per second squared east. You can put direction to it. So acceleration is the vector quantity. Now let's talk more about the difference between distance and displacement. So let's say a person traveled 13 meters east. And then he turns around and travels 4 meters west. We're going to add a negative sign to this because he's going towards the left. And just to review direction, north and south is along the y-axis. East and west, they're along the x-axis. So east is along the positive x-axis, west is along the negative x-axis. So given a situation, calculate the distance and the displacement of this individual that traveled here. Let's put a person. So the total distance that this person traveled is 17 meters. It's basically the sum of these two numbers. He traveled 13 meters east and then 4 meters west. So a total distance of 17 meters. Distance being scalar, it's always positive. Now there is an exception to this, specifically temperature. Depending on the units you're dealing with, even though temperature is a scale of quantity, you could have negative values. Especially for units such as Fahrenheit and Celsius. So some places can be negative 20 degrees Fahrenheit in temperature. Others negative 10 degrees Celsius. But on the Kelvin scale, there's no negative values. So there are some scalar quantities that do have negative values like temperature. So just be aware of that. Now going back to this problem. Now that we know the total distance traveled, what is the displacement of this particular individual? Now displacement can be positive or negative, dependent on the direction. If you were to combine these two, including the negative 4 value, it would be 13 minus 4, so the displacement is positive 9. Displacement is the final position minus the initial position. So displacement is the change in position. What this tells us is that this person ended up 9 meters east from where he started. And so let's illustrate this with a number line. So let's say that he started at the origin, position 0. He traveled 13 meters east. So because he's moving along the positive x-axis, we assign a positive value here. And then during the second part of his trip, he traveled 4 meters west. So 13 minus 4, he ends up at position 9. So the net result is that he traveled 9 meters east. And so that's his displacement. That's how far he traveled relative to his initial position. So hopefully this is example. gave you a good understanding between the difference of distance and displacement. So remember distance is a scalar quantity, it's always positive and displacement is a vector. It has both magnitude and direction. Now let's talk about the difference between speed and velocity. Remember speed is a scalar quantity but velocity is a vector quantity. Speed has magnitude only but velocity has both magnitude and direction Let's use s to describe speed S would represent the instantaneous speed that is how fast an object is moving at an instant of time But s bar represents the average speed Average speed is equal to the distance traveled divided by the time that was elapsed. Average velocity is equal to displacement over time. So therefore, speed is always positive. Velocity can be positive or negative, depending on the direction. So average velocity is basically the velocity calculated over a time interval. Now let's use an example to calculate speed, I mean average speed and average velocity. So let's say we have a particle, and this particle travels 100 meters east. And then it turns around and travels 150 meters west. And it does all of this in 5 seconds. Calculate the average speed and the average velocity of this particle over this 5 second time interval. So to calculate the average speed, first we need to calculate the total distance traveled. This particle traveled a total distance of 250 meters. It traveled 100 meters east and then 150 meters west. So if you add those two numbers, you get a total distance of 250 meters. Now it did this in 5 seconds. 250 divided by 5 is 50. So the average speed of this particle is positive 50 meters per second. Now, the average velocity is different because the displacement is different. Now, there's two ways in which we can calculate the displacement. We can add up the individual, the displacements for the individual segments of the problem. The displacement for the first part is 100 meters, and the displacement for the second part is negative 150. If we add 100 plus negative 150, we get a net displacement of negative 50. The other way in which you can calculate displacement is by taking the final position and subtracting it by the initial position. And it helps to draw a number line for this. So the particle started at position 0, and during the first part of it's traveled, It was at position 100, and then it traveled 150 meters west, so it ended at position negative 50. So its final position was negative 50 minus its initial position of 0, which would still be negative 50. I mean you could do it that way too, but I prefer to simply add the displacement of each individual part of the problem to get the final displacement. So the final displacement is negative 50 meters divided by a time of five seconds. So negative 50 divided by 5, the average velocity calculated is negative 10 meters per second. So as you can see, the average speed and the average velocity is not always the same. It's going to be the same if the person or the object travels in one direction. But if there's any change in direction, like when this particle decided to go west, that's when the average speed and the average velocity will be different. So keep in mind, the average speed doesn't have to equal the average velocity. Sometimes they're equal to each other, but it's not always the case. Now, for instantaneous speed, that is the absolute value of instantaneous velocity. The instantaneous velocity tells you the velocity at an instant of time. Whereas the average velocity basically tells you the average over an interval. So remember, average velocity is the displacement over time. And displacement is the final position minus the initial position divided by t. Now, instantaneous velocity, in order to calculate it, you need to use limits. So it's the limit as the change in time goes to zero. And it's basically the displacement over time, or the change in position over the change in time. So that's how you would calculate instantaneous velocity, which we really won't go over that in this video. But for those of you who want the formula, that's what it is. And whenever you see delta, this triangle, it represents the change of something. In this case, the change in position. So delta x is the final position minus the initial position. So basically, it's the displacement along the x-axis. But you can also have displacement along the y-axis. In this case, it will be the final position along the y-axis minus the initial position. So whenever you see the symbol D, you could describe it using distance or displacement. Now let's talk about some formulas that you need to be familiar with. When an object is moving with constant speed, typically this is the formula that you need to work with. d is equal to vt. So, in this equation, d can represent distance or displacement, depending on how you use the problem. v d You could use speed or velocity. It's going to work if you're dealing with constant speed. But understand this though. If you're using v as speed, then d is going to be the distance. If you're using v with reference to velocity, d is going to represent the displacement. Now for objects moving at constant speed, you need to know that the instantaneous velocity is the same as the average velocity because the velocity is not changing. So whether you use v or v bar, it'll have the same effect. Now let's talk about when objects are moving with constant acceleration. and by the way remember if you're using displacement displacement is the final position minus the initial position but if you're dealing with distance you don't have to worry about that so just keep in mind if you're using D as displacement or distance now for constant acceleration we have some formulas that we need to take into account For constant acceleration, d is equal to v-bar times t. And the reason for that is that v doesn't equal v bar when there's acceleration. Anytime there's acceleration, that means the velocity is changing. Acceleration tells you how fast the velocity is changing. And if the velocity is changing, then the instantaneous velocity and the average velocity for most of the time won't be the same. So thus we need to use the V bar. Now average velocity is the average of the initial velocity and the final velocity. So it's basically the sum of the initial and the final velocity divided by 2 or you can write it this way one-half the initial plus the final So if we were to replace V bar with this expression, we would get the displacement is equal to 1 half V initial plus V final times T. So remember, if you're using d as distance, then v initial is the initial speed, v final is the final speed. But if you're using d as displacement, v initial is your initial velocity, v final is your final velocity. So let's rewrite that equation here. This is one of those equations that you want to write in your list of equations if you have a test coming up. Now there's some other equations that we're going to add to this list. So we said that average velocity is the displacement, which is final position minus initial position, divided by t. Well, if you were to rearrange that equation, you'll get this one. Final position is equal to initial position plus the average velocity times t. you can also get it from this equation d is equal to V bar times T and D being the displacement is X final minus X initial and so if you were to move this to that side you would end up with the same equation. So there's many ways in which you can derive that equation. So that's the next one that you want to have in your list. Now, we said that average velocity is the displacement or the change of position divided by the time. Average acceleration is the change in velocity divided by time. the final velocity minus the initial velocity divided by T. So if you rearrange that equation, you get something similar to the one that we have above. V final is equal to V initial plus A T. Now there are some other equations that we can add to the list. Another one is V final squared is equal to V initial squared plus 2 times A D. And then there's this one. Displacement is equal to v initial t plus 1 half a t squared. And anytime you see the letter d, you can replace that with x final minus x initial. So if we were to substitute x final minus x initial for d, and then move x initial to the right side of the equation, we'll get this equation. Final position is equal to initial position. plus the initial T plus 1 half a t squared now because this is x we're dealing with motion along the x-axis But when you go into projectile motion, you can apply motion along the y-axis as well. So you might see this variation of this formula. y final equals y initial plus vy initial t. So that's initial velocity, but in the y direction, plus 1 half at squared, where a is like the gravitational acceleration in the y direction. So, just understand that you can apply these formulas in the x direction or in the y direction. So, it might be a lot to keep track of, but as you begin to work problems, the use of these formulas will begin to make more sense. But you may want to write these down for your reference. Now, before we work on a few example problems, there's something I do want to mention. When dealing with constant speed... You could use this formula. X final is equal to X initial plus VT. So you don't need to write V bar because V and V bar are the same when dealing with constant speed. But for constant acceleration... It's good to keep in mind that this represents average velocity. So it's better to write V bar instead of V so there's no confusion. Because these two, they're not necessarily the same when dealing with constant acceleration. So just be aware of that little detail. Let's start with this one. A bus is traveling at a constant speed of 40 meters per second. How many hours will it take to travel a distance of 200 miles? So what formula do we need to use? So we're given the speed which we can use as a symbol v or s. And we know the distance, which is 200 miles. The only formula that we could use, since the bus is moving at constant speed, is this one. D is equal to Vt. Now, before we use that formula, we need to take a look at the units. Here we have meters per second and here we have miles. And we want to find a time in hours. So we're looking for t. The units, they don't match. So before we can use the formula, we need to convert the units into an appropriate form. So what do you recommend that we need to do? We have the distance in miles and we want to find the time in hours. The best thing we can do is convert the speed from meters per second to miles per hour. Once we do that, then it will match with the unit hours and the unit miles. So we can now use that formula. So let's go ahead and convert 40 meters per second to miles per hour. So let's get an overview of how we're going to do this. Let's convert meters to kilometers, and then kilometers to miles. And then we're going to convert seconds to minutes, and then minutes to hours. So in order to convert miles to kilometers, you need to know the relationship between the two. One kilometer is equivalent to... 1000 meters. I meant to say meters instead of miles, but that's how you can go from meters to kilometers. Now, these units cancel. Now let's convert from kilometers to miles. So we need to put the unit kilometers on the bottom, miles on top, so that these units will cancel. You need to know that 1 mile is equal to 1.609 kilometers. So now we have the unit miles. Let's convert seconds to minutes. Since we have seconds on the bottom, we want to put seconds on top and then minutes on the bottom. 1 minute is equal to 60 seconds. So now we can cross out this unit. and now let's convert minutes to hours one hour is equivalent to 60 minutes and so we could cancel the unit minutes So now let's do the math. We're going to multiply by all the numbers that are on the top, but we're going to divide by the numbers on the bottom. So it's going to be 40 divided by 1,000. Take that result, divide it by 1.609, and then multiply that by 60, and then by another 60. For the velocity, or rather the speed, you should get 89.5, if you round it, miles per hour, which we can write mph. So that's how we can convert meters per second to miles per hour. Now let's go ahead and finish this problem. So let's use the formula d is equal to vt. In this case, D is going to represent the distance, which is 200 miles. V is going to be the speed, which is 89.5 miles per hour. And then now we can calculate T. So to get T by itself, we need to divide both sides by 89.5 miles per hour. 200 divided by 89.5 gives us this answer, 2.23 hours. Now looking at the units, we can see that the unit miles cancel, leaving behind the unit hours, which is what we want the final answer to be in. So when dealing with these problems, always check to make sure that the units match. If they don't match, you need to convert one unit into another. So just keep that in mind. Now let's move on to Part B of this problem. If the bus moved from a position that is 50 miles east of city XYZ to a position that is 90 miles west of city XYZ in 5 hours, what is the average velocity of the bus during that time interval? So, let's say this is city XYZ. So the bus was initially, let's say, at position A, which is 50 miles east of city XYZ. Actually, that's west. I'll take that back. Let's say city XYZ is at the origin, position 0. So the bus was 50 miles east. of the city, so it was initially here. So this is going to be x initial. That's the initial position of the bus. And then it moved to a position that is 90 miles west of that city. So this is going to be negative 90. That's the final position of the bus. So the bus is traveling west, which means that the average velocity should be a negative value. Now how can we calculate the average velocity? What formula do we need to use? It really helps to write down everything. We know that x initial is 50, x final is negative 90, and the units are in miles, and then we have the time, which is 5 hours. We want to calculate the average velocity. Well, we know that average velocity is displacement over time, and displacement is the change in position. X final minus X initial, whenever you're dealing with motion along the X axis, and then divided by the time. So the final position is negative 90, the initial position is positive 50, the time is 5 hours. so negative 90 minus positive 50 that is negative 140 so the change in position or displacement that is negative 140 miles and we're going to divide that by five hours So 140 divided by 5 is 28. Thus, the average velocity is going to be negative 28 miles per hour, which we can write as mph. So that's how we can calculate the average velocity for this particular problem.

Now the first thing we need to talk about is the difference between a scalar quantity and a vector quantity. A scalar quantity is something that has magnitude only, whereas a vector quantity has both magnitude and direction. So for instance, mass is a scalar quantity. You could say that a book has two kilograms of mass.

The two kilograms would be the magnitude of the mass of the book. But you wouldn't say the book has two kilograms of mass east. Direction wouldn't apply for mass.

So mass cannot be a vector quantity. Distance is a scalar quantity. And displacement is a vector quantity. Displacement, you can think of it as distance with direction. So let's say if a person traveled 15 meters, you're basically describing a person's distance because you didn't mention direction.

But let's say if someone traveled 15 meters east, now you're describing displacement. You're describing not only how far he traveled, but also where he traveled. So think of displacement as distance with direction. Now technically, displacement is really the change in the position of an object.

So along the x-axis, you could describe displacement as the final position minus the initial position. Speed is a scalar quantity. Speed describes how fast an object is moving.

Velocity is a vector quantity. Velocity is speed with direction. So if you were to say a person is traveling at 20 meters per second, you're describing the speed of the person.

This is the magnitude. But let's say if a person is moving at 30 meters per second north, you're describing velocity because You've mentioned his speed along with the direction. So you have both magnitude and direction, which makes it a vector quantity. Now what about temperature? Would you say that temperature is a scalar quantity, or would you describe it as a vector quantity?

So think of units of temperature. Let's say it's 80 degrees Fahrenheit outside. or 25 degrees Celsius.

You wouldn't say it's 80 degrees Fahrenheit north. Direction doesn't apply for temperature. So because you can't describe the direction of temperature, it's not relevant.

Temperature is a scale of quantity. You can only mention its magnitude, such as how hot it is, 80 degrees Fahrenheit, but you can't attach direction to temperature, which makes it scalar. Now acceleration, that is a vector quantity. Acceleration tells you how fast the velocity is changing.

And you could say a car is accelerating at 2 meters per second squared east. You can put direction to it. So acceleration is the vector quantity.

Now let's talk more about the difference between distance and displacement. So let's say a person traveled 13 meters east. And then he turns around and travels 4 meters west. We're going to add a negative sign to this because he's going towards the left. And just to review direction, north and south is along the y-axis.

East and west, they're along the x-axis. So east is along the positive x-axis, west is along the negative x-axis. So given a situation, calculate the distance and the displacement of this individual that traveled here.

Let's put a person. So the total distance that this person traveled is 17 meters. It's basically the sum of these two numbers. He traveled 13 meters east and then 4 meters west.

So a total distance of 17 meters. Distance being scalar, it's always positive. Now there is an exception to this, specifically temperature.

Depending on the units you're dealing with, even though temperature is a scale of quantity, you could have negative values. Especially for units such as Fahrenheit and Celsius. So some places can be negative 20 degrees Fahrenheit in temperature.

Others negative 10 degrees Celsius. But on the Kelvin scale, there's no negative values. So there are some scalar quantities that do have negative values like temperature. So just be aware of that. Now going back to this problem.

Now that we know the total distance traveled, what is the displacement of this particular individual? Now displacement can be positive or negative, dependent on the direction. If you were to combine these two, including the negative 4 value, it would be 13 minus 4, so the displacement is positive 9. Displacement is the final position minus the initial position.

So displacement is the change in position. What this tells us is that this person ended up 9 meters east from where he started. And so let's illustrate this with a number line.

So let's say that he started at the origin, position 0. He traveled 13 meters east. So because he's moving along the positive x-axis, we assign a positive value here. And then during the second part of his trip, he traveled 4 meters west. So 13 minus 4, he ends up at position 9. So the net result is that he traveled 9 meters east. And so that's his displacement.

That's how far he traveled relative to his initial position. So hopefully this is example. gave you a good understanding between the difference of distance and displacement. So remember distance is a scalar quantity, it's always positive and displacement is a vector. It has both magnitude and direction.

Now let's talk about the difference between speed and velocity. Remember speed is a scalar quantity but velocity is a vector quantity. Speed has magnitude only but velocity has both magnitude and direction Let's use s to describe speed S would represent the instantaneous speed that is how fast an object is moving at an instant of time But s bar represents the average speed Average speed is equal to the distance traveled divided by the time that was elapsed.

Average velocity is equal to displacement over time. So therefore, speed is always positive. Velocity can be positive or negative, depending on the direction. So average velocity is basically the velocity calculated over a time interval. Now let's use an example to calculate speed, I mean average speed and average velocity.

So let's say we have a particle, and this particle travels 100 meters east. And then it turns around and travels 150 meters west. And it does all of this in 5 seconds.

Calculate the average speed and the average velocity of this particle over this 5 second time interval. So to calculate the average speed, first we need to calculate the total distance traveled. This particle traveled a total distance of 250 meters. It traveled 100 meters east and then 150 meters west.

So if you add those two numbers, you get a total distance of 250 meters. Now it did this in 5 seconds. 250 divided by 5 is 50. So the average speed of this particle is positive 50 meters per second.

Now, the average velocity is different because the displacement is different. Now, there's two ways in which we can calculate the displacement. We can add up the individual, the displacements for the individual segments of the problem.

The displacement for the first part is 100 meters, and the displacement for the second part is negative 150. If we add 100 plus negative 150, we get a net displacement of negative 50. The other way in which you can calculate displacement is by taking the final position and subtracting it by the initial position. And it helps to draw a number line for this. So the particle started at position 0, and during the first part of it's traveled, It was at position 100, and then it traveled 150 meters west, so it ended at position negative 50. So its final position was negative 50 minus its initial position of 0, which would still be negative 50. I mean you could do it that way too, but I prefer to simply add the displacement of each individual part of the problem to get the final displacement.

So the final displacement is negative 50 meters divided by a time of five seconds. So negative 50 divided by 5, the average velocity calculated is negative 10 meters per second. So as you can see, the average speed and the average velocity is not always the same. It's going to be the same if the person or the object travels in one direction.

But if there's any change in direction, like when this particle decided to go west, that's when the average speed and the average velocity will be different. So keep in mind, the average speed doesn't have to equal the average velocity. Sometimes they're equal to each other, but it's not always the case. Now, for instantaneous speed, that is the absolute value of instantaneous velocity. The instantaneous velocity tells you the velocity at an instant of time.

Whereas the average velocity basically tells you the average over an interval. So remember, average velocity is the displacement over time. And displacement is the final position minus the initial position divided by t.

Now, instantaneous velocity, in order to calculate it, you need to use limits. So it's the limit as the change in time goes to zero. And it's basically the displacement over time, or the change in position over the change in time.

So that's how you would calculate instantaneous velocity, which we really won't go over that in this video. But for those of you who want the formula, that's what it is. And whenever you see delta, this triangle, it represents the change of something.

In this case, the change in position. So delta x is the final position minus the initial position. So basically, it's the displacement along the x-axis. But you can also have displacement along the y-axis. In this case, it will be the final position along the y-axis minus the initial position.

So whenever you see the symbol D, you could describe it using distance or displacement. Now let's talk about some formulas that you need to be familiar with. When an object is moving with constant speed, typically this is the formula that you need to work with. d is equal to vt.

So, in this equation, d can represent distance or displacement, depending on how you use the problem. v d You could use speed or velocity. It's going to work if you're dealing with constant speed.

But understand this though. If you're using v as speed, then d is going to be the distance. If you're using v with reference to velocity, d is going to represent the displacement. Now for objects moving at constant speed, you need to know that the instantaneous velocity is the same as the average velocity because the velocity is not changing.

So whether you use v or v bar, it'll have the same effect. Now let's talk about when objects are moving with constant acceleration. and by the way remember if you're using displacement displacement is the final position minus the initial position but if you're dealing with distance you don't have to worry about that so just keep in mind if you're using D as displacement or distance now for constant acceleration we have some formulas that we need to take into account For constant acceleration, d is equal to v-bar times t.

And the reason for that is that v doesn't equal v bar when there's acceleration. Anytime there's acceleration, that means the velocity is changing. Acceleration tells you how fast the velocity is changing. And if the velocity is changing, then the instantaneous velocity and the average velocity for most of the time won't be the same. So thus we need to use the V bar.

Now average velocity is the average of the initial velocity and the final velocity. So it's basically the sum of the initial and the final velocity divided by 2 or you can write it this way one-half the initial plus the final So if we were to replace V bar with this expression, we would get the displacement is equal to 1 half V initial plus V final times T. So remember, if you're using d as distance, then v initial is the initial speed, v final is the final speed.

But if you're using d as displacement, v initial is your initial velocity, v final is your final velocity. So let's rewrite that equation here. This is one of those equations that you want to write in your list of equations if you have a test coming up.

Now there's some other equations that we're going to add to this list. So we said that average velocity is the displacement, which is final position minus initial position, divided by t. Well, if you were to rearrange that equation, you'll get this one.

Final position is equal to initial position plus the average velocity times t. you can also get it from this equation d is equal to V bar times T and D being the displacement is X final minus X initial and so if you were to move this to that side you would end up with the same equation. So there's many ways in which you can derive that equation. So that's the next one that you want to have in your list. Now, we said that average velocity is the displacement or the change of position divided by the time.

Average acceleration is the change in velocity divided by time. the final velocity minus the initial velocity divided by T. So if you rearrange that equation, you get something similar to the one that we have above. V final is equal to V initial plus A T. Now there are some other equations that we can add to the list. Another one is V final squared is equal to V initial squared plus 2 times A D. And then there's this one.

Displacement is equal to v initial t plus 1 half a t squared. And anytime you see the letter d, you can replace that with x final minus x initial. So if we were to substitute x final minus x initial for d, and then move x initial to the right side of the equation, we'll get this equation.

Final position is equal to initial position. plus the initial T plus 1 half a t squared now because this is x we're dealing with motion along the x-axis But when you go into projectile motion, you can apply motion along the y-axis as well. So you might see this variation of this formula.

y final equals y initial plus vy initial t. So that's initial velocity, but in the y direction, plus 1 half at squared, where a is like the gravitational acceleration in the y direction. So, just understand that you can apply these formulas in the x direction or in the y direction. So, it might be a lot to keep track of, but as you begin to work problems, the use of these formulas will begin to make more sense. But you may want to write these down for your reference.

Now, before we work on a few example problems, there's something I do want to mention. When dealing with constant speed... You could use this formula.

X final is equal to X initial plus VT. So you don't need to write V bar because V and V bar are the same when dealing with constant speed. But for constant acceleration... It's good to keep in mind that this represents average velocity.

So it's better to write V bar instead of V so there's no confusion. Because these two, they're not necessarily the same when dealing with constant acceleration. So just be aware of that little detail. Let's start with this one.

A bus is traveling at a constant speed of 40 meters per second. How many hours will it take to travel a distance of 200 miles? So what formula do we need to use? So we're given the speed which we can use as a symbol v or s.

And we know the distance, which is 200 miles. The only formula that we could use, since the bus is moving at constant speed, is this one. D is equal to Vt. Now, before we use that formula, we need to take a look at the units. Here we have meters per second and here we have miles.

And we want to find a time in hours. So we're looking for t. The units, they don't match. So before we can use the formula, we need to convert the units into an appropriate form. So what do you recommend that we need to do?

We have the distance in miles and we want to find the time in hours. The best thing we can do is convert the speed from meters per second to miles per hour. Once we do that, then it will match with the unit hours and the unit miles.

So we can now use that formula. So let's go ahead and convert 40 meters per second to miles per hour. So let's get an overview of how we're going to do this. Let's convert meters to kilometers, and then kilometers to miles. And then we're going to convert seconds to minutes, and then minutes to hours.

So in order to convert miles to kilometers, you need to know the relationship between the two. One kilometer is equivalent to... 1000 meters.

I meant to say meters instead of miles, but that's how you can go from meters to kilometers. Now, these units cancel. Now let's convert from kilometers to miles. So we need to put the unit kilometers on the bottom, miles on top, so that these units will cancel.

You need to know that 1 mile is equal to 1.609 kilometers. So now we have the unit miles. Let's convert seconds to minutes. Since we have seconds on the bottom, we want to put seconds on top and then minutes on the bottom.

1 minute is equal to 60 seconds. So now we can cross out this unit. and now let's convert minutes to hours one hour is equivalent to 60 minutes and so we could cancel the unit minutes So now let's do the math.

We're going to multiply by all the numbers that are on the top, but we're going to divide by the numbers on the bottom. So it's going to be 40 divided by 1,000. Take that result, divide it by 1.609, and then multiply that by 60, and then by another 60. For the velocity, or rather the speed, you should get 89.5, if you round it, miles per hour, which we can write mph.

So that's how we can convert meters per second to miles per hour. Now let's go ahead and finish this problem. So let's use the formula d is equal to vt.

In this case, D is going to represent the distance, which is 200 miles. V is going to be the speed, which is 89.5 miles per hour. And then now we can calculate T.

So to get T by itself, we need to divide both sides by 89.5 miles per hour. 200 divided by 89.5 gives us this answer, 2.23 hours. Now looking at the units, we can see that the unit miles cancel, leaving behind the unit hours, which is what we want the final answer to be in.

So when dealing with these problems, always check to make sure that the units match. If they don't match, you need to convert one unit into another. So just keep that in mind.

Now let's move on to Part B of this problem. If the bus moved from a position that is 50 miles east of city XYZ to a position that is 90 miles west of city XYZ in 5 hours, what is the average velocity of the bus during that time interval? So, let's say this is city XYZ.

So the bus was initially, let's say, at position A, which is 50 miles east of city XYZ. Actually, that's west. I'll take that back. Let's say city XYZ is at the origin, position 0. So the bus was 50 miles east. of the city, so it was initially here.

So this is going to be x initial. That's the initial position of the bus. And then it moved to a position that is 90 miles west of that city. So this is going to be negative 90. That's the final position of the bus.

So the bus is traveling west, which means that the average velocity should be a negative value. Now how can we calculate the average velocity? What formula do we need to use?

It really helps to write down everything. We know that x initial is 50, x final is negative 90, and the units are in miles, and then we have the time, which is 5 hours. We want to calculate the average velocity.

Well, we know that average velocity is displacement over time, and displacement is the change in position. X final minus X initial, whenever you're dealing with motion along the X axis, and then divided by the time. So the final position is negative 90, the initial position is positive 50, the time is 5 hours.

so negative 90 minus positive 50 that is negative 140 so the change in position or displacement that is negative 140 miles and we're going to divide that by five hours So 140 divided by 5 is 28. Thus, the average velocity is going to be negative 28 miles per hour, which we can write as mph. So that's how we can calculate the average velocity for this particular problem.

Transcript for:Understanding Kinematics: Motion Principles

Transcript for:
Understanding Kinematics: Motion Principles