In this video, we're going to talk about acceleration. But before we do that, let's talk about the difference between velocity and acceleration. Both quantities are vectors. They both have magnitude and direction.
But what is the difference between velocity and acceleration? Velocity tells you how fast... An object's position is changing with respect to time. So basically, velocity is the displacement, which is the change in position over time.
So V is equal to the displacement is the final position minus the initial position divided by the elapsed time. Now, as t approaches 0, this expression is, or becomes, rather, the instantaneous velocity. The previous equation represents the average velocity.
So if you want to find the average velocity, simply use this equation. It's the change in position over time, or displacement over time. So that will give you the average velocity.
Whereas the instantaneous velocity, you can use the same formula, but when t becomes very, very small, it will give you the instantaneous velocity. Now let's say if you have a car that's moving at 30 miles per hour. traveling east. So this expression, the magnitude and the direction, that represents the velocity of the vehicle. So every hour the car travels a distance of 30 miles.
So in two hours, it's going to travel 60 miles. Three hours, 90 miles. So speed tells you how fast the distance is changing. Velocity tells you how fast the displacement is changing.
So let's say if a car, or let's say if an object's moving at a speed of 8 meters per second. Every second, the distance is changing by 8 meters. So after one second, the distance traveled is 8 meters.
Two seconds, 60 meters. 3 seconds, 24 meters. So, that's the main concept between speed and velocity.
It tells you how fast the distance or the displacement is changing every second. So, if a car is moving at a speed of 15 meters per second, every second the distance is changing by 15. meters. So to sum up what we just considered, speed tells you how fast the distance is changing. Velocity tells you how fast the displacement is changing.
Acceleration tells you how fast the velocity is changing. So acceleration, this is average acceleration, it's equal to the change in velocity, you can say delta v if you want to, over time. That'll give you the average acceleration. If you want to find an instantaneous acceleration, as t approaches zero, when t becomes very small, that is the change in time, This expression will give you the instantaneous acceleration.
But if it's over a significant time interval, then that equation will give you the average acceleration. So just as speed tells you how fast the distance is changing, acceleration tells you how fast the velocity is changing. So let's say if the acceleration is 8 meters per second squared. So that means every second the velocity is changing by 8 meters per second. So in one second, let's say, let's make a table.
Let's say this is time and velocity. After one second, the speed will be 8 meters per second. After two seconds, the velocity will be 16. Three seconds, 24. And then four seconds, 32. So, every second, the velocity is increasing by 8. So, that's the main idea behind acceleration. It tells you how fast the velocity is changing. Now starting from this equation, there's another equation that you might find useful.
Let's multiply both sides by t. So a t is equal to v final minus v initial. Now let's add v initial to both sides. So v initial plus a t is equal to v final. So anytime you want to calculate the final speed, For motion or any object that's undergoing accelerated motion, you can use this equation.
The acceleration has to be constant, by the way, for this to work. So when an object is moving with constant acceleration, you can find the final speed given the initial speed, the acceleration, and the time. Now let's work on some problems.
A car accelerates from 15 mps to 45 mps in 5 seconds. Calculate the average acceleration of the car. Now what I'd like to do is I'd like to make a list of what I have and what I need to find.
So, the initial speed in this problem is 15 m per second. The car accelerates from that speed to a final speed of 45 m per second. The time in between those two points is 5 seconds.
Our goal is to calculate the average acceleration. So the formula that we could use is this one. Acceleration is equal to V final minus V initial divided by T.
The final speed is 45. The initial speed is 15. And the time is 5. So the chain... in speed is 30 meters per second and we're going to divide that by 5 seconds 30 divided by 5 is 6 so the acceleration is 6 meters per second squared now let's understand this problem so let's make a table so this is the acceleration Let's make a table between time and velocity. At t equals 0, the initial speed is 15 meters per second. Now keep in mind, the acceleration tells us how much the speed changes or the velocity changes every second.
So one second later, it's going to be 6 meters per second higher. So 15 plus 6 is 21. Whenever the acceleration is positive, the velocity is increasing. 2 seconds later, it's going to be at 27. So every second, the speed is going to increase by 6 meters per second.
So at 3 seconds, it's going to be 33. And at 4 seconds, it's going to be 39. and five seconds later, it's going to reach to our final speed of 45 ms. So I want you to understand the concept of acceleration. And just think of it as how fast the speed changes, or how fast the velocity changes, every second. Number two, a truck accelerates from 25 kilometers per hour to 45 kilometers per hour in 40 seconds. Calculate the average acceleration of the vehicle in kilometers per hour per second and in meters per second squared so go ahead and take a minute and we're going to problem so just like before we're going to make a list of what we have and what we need to find so what does the 25 kilometer per hour value represent so that is the initial speed of the vehicle now we know that the final speed has to be 45 kilometers per hour And the time in between these two events is 40 seconds.
So now we could use the same formula to calculate the average acceleration. So the average acceleration is the change in the velocity divided by the time. The final speed is 45. The initial speed is 45. is 25 and the time is 40 so 45 minus 25 that gives us a change of 20 and the units is kilometers per hour and let's divide that by 40 seconds Well, 20 divided by 40 is the same as 2 divided by 4, which is 1 half, or 0.5. Now, notice what the units for this acceleration will be.
We have kilometers, hours, divided by seconds. So, the units is going to be kilometers per hour per second. So, that's our answer.
That's the acceleration. Now let's focus on the second part of the problem. We need to convert this unit to meters per second squared, which is the most common unit you'll see for acceleration.
So how should we divide this on the first fraction? 0.5 has to go on top and kilometers. It's 0.5 kilometers per hour per second, so the second two units have to go in the bottom. Now, what we need to do is change kilometers to meters. One kilometer.
is equivalent to a thousand meters. And we need to write it in such a way so that the unit kilometers will cancel. Now all we need to do is convert this hour into seconds. So one hour is equal to 60 minutes.
and one minute is equivalent to 60 seconds so the unit hours cancel and the unit minutes cancel so now we have meters per second per second or meters per second squared 0.5 times 1000 is 500. And we're going to divide that by 60, which is about 8.33, and divide that by 60 again. So the acceleration of this vehicle is 0.138, repeating, meters per second squared. So that's the answer. So make sure you know how to convert from one unit of acceleration into another. Here's another problem that we can work on.
A car accelerates from rest at a constant rate of 3.5 meters per second squared. What is the speed of the car 12 seconds later? So, pause the video and work on this problem.
See if you can find the answer. Now what variables do we have in this problem? What is the initial speed and what is the final speed? What is the time and what is the acceleration as well?
Now, whenever you hear the word rest, that tells you that the speed is zero. The car accelerates from rest. That means it's accelerating from an initial speed of zero. Our goal is to find the final speed of the vehicle.
So that's what we're looking for. The time is 12 seconds, and the acceleration is 3.5 meters per second squared. So what equation do we need to use in order to calculate the final speed?
So we need to use this formula. The final speed is equal to the initial speed plus the product of the acceleration and the time. The initial speed is 0, the acceleration is 3.5, and the time is 12. So all you need to do is multiply 3.5 times 12, and that will give you the final speed of 42 meters per second.
Number four, a bus accelerates from an initial speed of 12 meters per second at a constant rate of 1.2 meters per second squared. What is the final speed of the bus after 15 seconds? So just like before, let's make a list of the variables that we have and a variable that we need to find. So we're given the initial speed, which is 12 meters per second.
And looking at the units meters per second squared, we can see that that value represents the acceleration. And if you see the unit seconds, you know that represents time. Our goal is to find the final speed of the bus.
So we need to calculate Vf. Therefore, we can use the same exact equation as the one we used in the last problem. So Vf is equal to V initial plus 80. The initial speed is 12. The acceleration is 1.2. And the time is 15 seconds.
So first, let's multiply 1.2 by 15. 1.2 x 15 is 18 and 12 plus 18 is 30. So the final speed is 30 meters per second. Number five. A sports car driver traveling at 95 miles per hour slams the brakes and comes to rest in four seconds. Calculate the average acceleration of the current in meters per second squared. So let's think about this.
If we want to find the acceleration in meters per second squared, We need to convert the speed from miles per hour to meters per second. It's not like the second problem where we had to get the answer in terms of kilometers per hour per second. If we only needed to get the answer in meters per second squared, I would have changed the units from the beginning.
But in this case, we need to get it in meters per second squared. So let's begin by converting miles per hour to meters per second. So we have 95 miles per hour.
One hour contains 60 minutes. and each minute contains 60 seconds. Now how can we convert miles to meters?
One way we can do it is we can convert miles to kilometers and then kilometers to meters. So we need to know the conversion. If I recall one kilometer is 0.6214 miles. And we know that 1 km is also 1000 m.
So as you can see, the unit miles cancels. Hours cancels as well. The same is true for minutes.
And finally, km. So we're left with m per second. now let's do the math 95 divided by 60 divided by another 60 divided by point 6214 times 1000 is about 42.47.
So that's the initial speed of the vehicle. It's 42.47 meters per second. Now, what is the final speed of the vehicle? If the driver slams the brakes, he's going to come to rest eventually.
So the final speed is 0. The time, we can see it's 4 seconds. And our goal is to find the acceleration. So let's use this formula a is equal to V final minus V initial divided by T so it's 0 minus 42 point 47 Divided by 4 So negative 42.47 divided by 4, that's about negative 10.6 meters per second squared. Now why is the acceleration negative? The speed is decreasing.
Anytime the velocity is decreasing, which is the magnitude of speed. Speed is the magnitude of velocity. But anytime the velocity is decreasing, the acceleration will be negative. And the car is slowing down. It's coming to a stop.
And so that's why you should have a negative average acceleration. So that's what it means.