A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at 2500 m/s . It is loaded with 600 kg of fuel, which it burns in 30 s. What is the rocket's speed 10 s, 20 s, and 30 s after launch?

Video transcript

Hey, everyone in this problem, a space shuttle is moving at 400 m per second in a region where the effect of gravitation is negligible. It accelerates at time T equals zero by consuming all of its fuel at a steady rate. In 40 seconds. The exhaust gasses are ejected at a constant speed of 1250 m per second. The empty shuttle mass is 1100 kg and the fuel mass is kg. We're asked to find the shuttle speed K at T equals seconds and at T equals 40 seconds. Now we're given four answer choices, options A through B or sorry A through D each of them containing different speeds or both part one and two of this problem in meters per second. And we're gonna come back to those when we're done working through this problem. No, let's get started on this problem. OK. We're told a lot of information. But one thing that's important is that this gravitation is negligible. OK? If we know that gravitation is negligible, then the equation for the speed recall is gonna be V is equal to V, not MVEX, the exhaust speed multiplied by law in the natural log of M divided by M. All right. So this is a lot of variables in this equation. Let's work through them. One by one. V naught is gonna be the initial speed. OK? We know the initial speed we were told that in the problem. So V is gonna be 400 meters per second. All right. What about this exhaust speed? Our exhaust speed. VX we're told a again 1250 m per second. So we have that value as well. OK. And notice that we've said that this is positive. OK. This variable vex is the exhaust speed. So we don't include a negative here. This is always a positive value and then we have the negative in our equation and you have to watch some textbooks. Some professors will consider that in a different way. Sometimes this will be the exhaust velocity. You'll need to include the negative in the actual value and the equation will have a positive. OK. So just watch out for that. All right. Now M is gonna be the mass at whatever time point we're at. OK. So we're gonna calculate that later for each part of this question. But M not, is gonna be the initial mass. And so that total initial mass and not is gonna be the mass of the rocket plus the mass of the fuel. OK. At the beginning, this rocket ship is full of fuel. And so we have that total mass of the rocket, 1100 kg plus the total mass of fuel kg or an M not value of kg. All right. So we have V knot, this exhaust velocity M knot. And we've said that M is gonna depend on the time. So let's get started with part one at time. T equals 20 seconds. Ok? So this is gonna be at T equals 20 seconds. And the question is what mass is remaining? Ok. So the max M and I'm gonna call it M 20. Just so we know that we're talking about at 20 seconds, it's gonna be equal to one. Oh, it's gonna be the initial mass. OK? We're starting with some initial laughs and then we're subtracting some fuel that we're burning, which we're gonna call DM by DT. A multiplied by the time T, what this equation is really saying this second part, this DM by DT is how much mass we're losing every single time point. We're gonna multiply that by the amount of time that we've been moving for, that's gonna give us the total amount of fuel that is burned. And then we're gonna subtract that from the initial mass because that fuel is no longer part of our mass. All right. Now, what is this D MDT thing? Ok. How do we know what that value is? Well, DM, by DT, it's gonna be the change in mass and the change in time, we're told that this space shuttle consumes all of its fuel in 40 seconds. And what's important here is that it's at a steady rate, ok. So it is gonna consume 400 kg of fuel in 40 seconds, meaning the rate that it's consuming fuel is 400 kg divided by 40 seconds. And this just gives us a rate of 10 kg per se. All right, now that we know that we can get back to our equation for the mass of 20 seconds. Ok. This is gonna be that total mass of 1500 kg minus the burn rate or the amount of fuel that we're getting rid of at each time point. So 10 kg per second multiplied by the total number of seconds, which is 20 per part. What? So we have 1500 kg minus 10 kg per second, multiplied by 20 seconds. We can see that the unit of seconds will divide in the second half of this equation. We're gonna be left with just kilograms, which is great. Yeah, we get that, this mass is going to be equal to 1300 kg. All right. So now that we have our mass M, we can actually calculate our speed. Ok. So our speed V for call is gonna be the initial speed V knot which is 400 m per second minus the exhaust speed, 1250 m per second, multiplied by the natural law of the fuel at or sorry, the mass at 20 seconds, which we've just found to be 1300 kg divided by the initial mass of 1500 kg. And when we work all of this out on our calculator, we get this speed of 0.876 m per second. And that makes sense. Ok. This speed is higher than our initial speed. Our rocket ship has exhausted some of that fuel to burn that fuel to increase the speed. Ok. So we have this higher speed, which makes sense all righty. So we're done part one, we're gonna move on to part two and it's gonna be very, very similar. And actually part two is gonna be a little bit simpler. And why is part two gonna be simpler? Well, at 40 seconds, we're told that all of the fuel is consumed, ok? If all of the fuel is consumed, then we don't need to use this fuel burn rate to calculate the mass. That's low. We know that the mass is just going to be the mass of the rocket ship which is 1100 kg and we don't have that massive fuel anymore. And so our speed V is gonna be equal to 400 meters per second minus the exhaust velocity, 1250 m per second, multiplied by the natural log of 1100 kg. The mass that's currently left divided by kg, the initial max and when we work all of this out on our calculator, we get that the speed V at 40 seconds is about 787. 3 6 6 m per second. And once again, this is higher than even that speed at 20 seconds. Which makes sense. We were still burning fuel. We were still using that fuel to increase our speed and push our rocket ship further. All right. So we're done with this problem. We've done some rocket science. Well, that's cool. Let's go up and check our answer traces. Now we found that the speed at 20 seconds. Ok. Rounding to the nearest meter per second is about 579. So option A has 442. We can eliminate that. Option D has 788 for that first part. So we can eliminate that. So we're looking at option B or C. Ok. 579 m per second at 20 seconds. What it comes down to is now the speeding 40 seconds which we found to be about 788. So this is gonna correspond with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.