Question Video: Using the Power of the Engine and the Speed to Find the Magnitude of the Resistance Force Mathematics

A train of mass 170 metric tons is moving along a horizontal section of track at a constant speed of 60 km/h. Given that the engine’s power output is 410 hp, find the magnitude of the resistance, 𝑅, to the train’s motion per metric ton of the train’s mass.

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Video Transcript

A train of mass 170 metric tons is moving along a horizontal section of track at a constant speed of 60 kilometers per hour. Given that the engine’s power output is 410 horsepower, find the magnitude of the resistance 𝑅 to the train’s motion per metric ton of the train’s mass.

So what we have is our train, and it’s moving along a track at constant speed. And we know that that speed is 60 kilometers per hour. And we know that the current power output to maintain that speed is 410 horsepower. And we also know there’s gonna be a locomotive force. And what you would think is, hold on, if there’s a force, would that mean that the train would be accelerating? However, what we also know is that there are opposing forces. And it’s these opposing forces, well, the magnitude of resistance, that we’re looking to find in this question.

But what we do know is that the magnitude of resistance is going to be equal to our force, so our electromotive force. And that’s because the train is traveling at a constant speed, so there is no acceleration. So therefore, what we can do is use our formula for power. And that is that power is equal to the force multiplied by the speed. However, if we want to find the force, and ultimately the resistant force, then we can rearrange our formula to force is equal to power divided by speed.

So you might think at this point, okay, great, we just plug in our values. Well, this isn’t the case. And that’s because our units are not in the correct units. So what we want to do is convert our units to the standard SI units. So I’m gonna start with power. And what we know is that one horsepower is equal to 735 watts. So therefore, our power is gonna be equal to 410 multiplied by 735, and that’s gonna be watts, which is gonna give us a power of 301,350 watts. Okay, great. So that’s our power dealt with.

So then next, we can concentrate on the speed. And we know that one kilometer per hour is equal to one over 3.6 meters per second. So therefore, we know that the speed is going to be equal to 60 multiplied by one over 3.6 meters per second, which is the same as 60 divided by 3.6 meters per second, which is gonna be equal to 16 and two-thirds meters per second. I’ve written it here as a fraction just to maintain accuracy as we’re gonna be using these values later in the calculation.

So therefore, we can now use our formula to calculate the force. And this is gonna be equal to 301,350, so that’s our power, divided by 16 and two-thirds, which is our speed, which is gonna give us a force of 18,081 newtons. Okay, great. But what we already said is that this would be the same as the resistant force. However, in this question, what we want to find out is not just the resistive force, but in fact, the magnitude of the resistance to the train’s motion per metric ton of the train’s mass.

Well, what we know from the question is the mass is equal to 170 metric tons. So therefore, we can say that the magnitude of the resistance 𝑅 to the train’s motion per metric ton of the train’s mass is going to be equal to, then we’re gonna have the force that we calculated divided by the train’s mass, which is gonna be equal to 18,081 newtons over 170 metric tons. I’ve kept the units in on this calculation just so we can see what the units of our answer are going to be. We can see that it’s gonna be newtons per metric ton. So therefore, this is gonna be equal to 106.36 newtons per metric ton. And we’ve rounded it here to two decimal places.

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