Video Transcript
The power of an aeroplane’s engine is 420 horsepower. It is flying against air resistance whose magnitude is proportional to the square of its speed. Given that the maximum speed of the plane is 441 kilometers per hour, determine the magnitude of the air resistance when its speed is 294 kilometers per hour.
Okay, so in this question what we have is an aeroplane with an engine that has a power of 420 horsepower. And we know that its maximum speed is 441 kilometers per hour. And we’re gonna consider this first of all because we’re actually gonna imagine that our aeroplane is actually traveling at its maximum speed. And what we’ll also have is a force, and that force will be a force in the direction of travel, so a locomotive force to the aeroplane.
However, what we know about this if we’re looking at the point when the aeroplane is traveling at its maximum speed is that this force is going to be equal to the air resistance or resistive force, which I’ve called 𝐹 sub R. And this is because there is no acceleration. And this comes from Newton’s first law, which says that an object stays in the same state of motion unless a resultant force acts upon it. So therefore, our resultant force must be equal to zero, so the forces must be equal. So therefore, what we can do is use our formula for power, which is power is equal to force times speed, to help us calculate our force and ultimately our air resistance. But what we can do is rearrange our formula. And we’re gonna get our force is equal to the power over the speed.
Well, we might think, “Well, great. All we need to do is plug in our values then because we’ve got up our power and we’ve got our maximum speed.” However, this wouldn’t be the case because what this would give us our force in would be horsepower per kilometer per hour, which is not the units that we’re looking for. So first of all, what we need to do is convert into our SI base units. So, if we take a look at power first, we know that one metric horsepower is equal to 735 watts. So, our power is going to be equal to 420 multiplied by 735 watts, which is gonna give us 308,700 watts.
And next, if we move onto the speed, then what we’ve got as our conversion is that one kilometer per hour is equal to one over 3.6 meters per second. So therefore, our speed is gonna be equal to 441 multiplied by one over 3.6 meters per second, which is the same as 441 divided by 3.6 meters per second, which will be equal to 122.5 meters per second. Okay, great. So we’ve now got our power and our speed in the correct units. So let’s use our formula to find out our force.
Well, our force is gonna be equal to the power over the speed. So therefore, it’s gonna be equal to 308,700 over 122.5. And this’s gonna be equal to 2,520, and then the units here are gonna be newtons. Okay, great. So we found our force at the maximum speed of the aeroplane. However, what do we want to do now? Well, what we want to do now is determine the magnitude of the air resistance when its speed is 294 kilometers per hour. But how are we gonna do this? Well, let’s clear some space and work out the next part of the question.
So then, what we can recall is that we said that the air resistance was gonna be equal to the force, remember, at the maximum speed. So therefore, we can say that the air resistance is equal to 2,520 newtons at 122.5 meters per second. And that’s because that’s what our speed was when converted into meters per second. So what do we want to do now? Well, as we said, we want to determine the magnitude of the air resistance when the speed is 294 kilometers per hour. So what this’s gonna turn into now is in fact a proportionality problem. And that’s because we’re told in the question that the magnitude of the air resistance is proportional to the square of its speed.
So what we can do is set up an equation. And that is that 𝐹 sub R is equal to 𝑘𝑣 squared. So, we’ve got here our 𝑣 squared. So that’s our speed squared. And we’ve got our air resistance. But then we’ve also introduced 𝑘, but what’s this? Well, what we can recall is that this is the proportionality constant. And what we want to do is find out what this is because this will then allow us to find out what the magnitude of the air resistance is when the speed is 294 kilometers per hour. But how can we do this? Well, we can do this because we’ve got two bits of information that we can put into this formula. And that’s because we know that 2,520, which was the air resistance, is equal to 𝑘 multiplied by and then our speed squared, 122.5 squared. So therefore, if we rearrange this, we found our proportionality constant because 𝑘 is equal to 2,520 over 122.5 squared.
Now, we could work this out, but what we’re gonna do is leave it in this form just to maintain accuracy for the rest of our calculations. So now, what we can do is in fact substitute this back into our formula to create a new formula for air resistance. And this is that the air resistance is equal to 2,520 over 122.5 squared multiplied by our speed squared. Well, we can think, “Great, let’s just plug in our values now and work out what the air resistance is when the speed is 294 kilometers per hour.” But no, once again, we’ve gotta remember to convert to our SI units, so we’re gonna use one of our conversion factors again. And that is that one kilometer per hour is equal to one over 3.6 meters per second.
So therefore, our speed is gonna be equal to 294 multiplied by one over 3.6 meters per second, which is gonna be equal to 81 and two-thirds meters per second. And we’re keep it as a fraction just to maintain accuracy. So therefore, our air resistance is gonna be equal to 2,520 over 122.5 squared multiplied by 81 and two-thirds squared. And this will give us the final answer for the magnitude of the air assistance when the speed is 294 kilometers per hour, and that is 1,120 newtons.
