A Porsche 944 Turbo has a rated engine power of 217 hp. 30% of the power is lost in the engine and the drive train, and 70% reaches the wheels. The total mass of the car and driver is 1480 kg, and two-thirds of the weight is over the drive wheels. (b) If the Porsche accelerates at aₘₐₓ, what is its speed when it reaches maximum power output?

Video transcript

Hey everyone. So this problem is working with power. Let's see what it's asking us. A pickup truck's engine delivers a maximum power of hp. The truck is loaded to a total mass of kg. Inefficiencies in the engine and transmission dissipate 20% of the generated power while 80% is successfully delivered to the drive wheels, the truck's mass is distributed. So the drive wheels carry 3/5 of the total mass when the truck accelerates at a maximum acceleration With a coefficient of static friction or us of 0.9 determine the speed attained at maximum power output. They're giving us a hint here to draw a free body diagram of the truck. Our multiple choice answers are a 7.51 m per second. B 5.41 m per second. C 1.35 m per second or D six point oh one or 6.1 m per second. Ok. So the hint was great. I like to start with free body diagrams pretty much whenever I don't know where else to start. So this is a good starting point, we have the truck, we know that the force acting on the truck in the positive direction is our static friction. And that hint was given to us because they are a second hint, right? They gave us that coefficient of static friction. We have the normal force acting in the positive Y direction and then the weight acting in the negative Y direction. The next step is to recognize they're giving us power and they're asking us to determine the speed at the maximum power output. So we can recall that our power is given by force multiplied by velocity. Our, we are looking for velocity so we can rearrange that to be velocity equals power divided by force. So that power term is pretty straightforward. And then we will talk through how to get that force term to solve for our final equation for speed or velocity. OK. So the power at the drive wheels We're told is 80 of the maximum power. So that's 0.8 multiplied by hp. And then we need to recall The conversion factor between horsepower and watts and that's 746 watts per horsepower. And we can plug that in and we get our power at the drive wheel of 7.162 Times 10 to the 4th watts. So solving for that power was pretty straightforward, given the information from the problem. Now we need to solve for the force. So the, some of the forces in the X direction, we can recall Newton's second law, which tells us that the, some of the forces equals mass multiplied by acceleration. And we'll do that in both the X and Y directions. So the, some of the forces in the X direction, the only, the only force is our friction or, or static friction force and that equals mass multiplied by acceleration. And then our static friction force itself is given by us N. And so this is interesting because really what we're after here is this static friction force, this F FS of S so we only need to solve or us multiplied by N. We're not really worried about the second half of that. The second part of that equation, we do need to find that N our normal force. And so we will use our Newton's second law in the Y direction to find that. So let's talk through that the, some of the forces in the Y direction that's going to be our normal force minus our weight is equal to mass multiplied by acceleration. There's no acceleration in the Y direction. So that equals zero, this whole term goes to zero. So our normal force is equal to our weight. However, we need to recall that the problem tells us that at the drive wheels, which is what we're focused on here. It, the drive wheels take 3/5 of the load. So our weight is given by mass times gravity, we can recall that. But instead of the full mass, it will only be 3/5 of the mass or 3/5 can be rewritten as 0.6. So our normal force is equal to 0.6 multiplied by mass multiplied by gravity. We'll plug that back in over here in our force is equal to Mute. S multiplied by 0. multiplied by mass multiplied by gravity. And now we know all of these values we can plug them in and solve for our force. So our coefficient of static friction, the us was given to us as 0.9 Multiplied by 0.6, multiplied by the mass kg Multiplied by gravity constant, which we can recall is 9.8 m/s squared. We can plug that in to our calculator and get a force of 1.19, 1 times 10 to the 4th Jews. And now we can go back up to that first equation. We have our power. So velocity equals power divided by force. We have our power 7.16, 2 Times 10 to the 4th watts Divided by that force we just saw for 1.19, 1 Times 10 to the 4th jewels And that equals 6. meters per second. And so that's the answer to this problem. And that aligns with answer choice D. So that's all we have for this one, we'll see you in the next video.