A 200 kg weather rocket is loaded with 100 kg of fuel and fired s...

Question

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s² for 30 s, then runs out of fuel. Ignore any air resistance effects.

a. What is the rocket's maximum altitude?

Accepted Answer

Hey everyone, welcome back in this problem. We're told that the vertical ascent phase of a 25 100 kg space shuttle last 26 seconds during this phase. The engine of the space shuttle provides a constant acceleration of 15 m per second squared At T equals 26 seconds the engine fails. We're told to neglect the effect of resistive forces and were asked to determine the maximum height attained by the space shuttle. Now we're given two different phases. Okay? We have the vertical ascent phase where the engine is providing acceleration and then we have the phase where the engine fails and so we're gonna need to break this problem up into two. So, during the vertical ascent phase, the initial speed During the ascent phase, which we're going to say is Phase one 0m/s. Okay, we're going to launch a rocket or a space shuttle from rest. So, the space shuttle starts at rest, it gets accelerated and it gets accelerated at 15 m/s squared the final speed of that rocket. When the engine stops working, we don't know that, But we do know that this entire thing lasts 26 seconds. Okay, this portion of the space shuttle launch where we have constant acceleration. Now, what we want to figure out is the height, we want to know the maximum height. So we want to figure out delta Y the height attained in this phase K. The vertical ascent phase, and this should say vertical ascent. All right now, in the second part of the problem, we don't know any information about the initial speed. We know the acceleration is going to be due to gravity. We aren't given information about the time or anything else. So we need to find some more information for the second phase. And what you'll notice is that the final speed in the first phase is going to be equal to the initial phase or the initial speed in the second phase. Okay, so whatever speed we have when the engine fails, it's gonna be the speed that we have Going into the 2nd phase. Alright, so we need to determine VF1, the final speed in this vertical ascent phase in order to find Some information about that second phase. But we also need to find Delta Y1 because that's going to relate to that maximum height that we're actually trying to find. So, let's start with this speed and this is a one and T one. So the final speed V F one is going to be equal to the initial speed. V not one plus a one times T one. Okay, we chose the U. Am equation. That doesn't include delta Y one to start And we get that v. F one is equal to zero meters per second plus 15 m per second squared times 26 seconds, which gives 390 m/s for that final speed during the vertical ascent phase. All right, so we know this final speed in the vertical ascent phase. Now let's find Delta Y. one. The height that we reach during that phase. Now, we're going to choose the U. A. M. Equation. And at this point we know V not V. F. We know a. And we know t. We can choose any of the um equations that contain the value. We're looking for delta Y one. Okay, So we're gonna use this equation. Delta Y one is equal to v nought T plus one half a T squared. And these are all the not one T one A one T one. Because we're talking about this first phase. And so this is going to be equal to zero m per second times T. That's just going to give us zero, Then we have 1/2 times a m per second squared times t squared, 26 seconds, all squared. And when we work this out, we're going to get 5070 m. We're gonna put a red box around this because this is going to relate to our final answer, and we're gonna come back to this later. Alright, So, we've done the vertical ascent phase, when the engine's working, we figured out how high it gets and we figured out the speed that the space shuttle is going before the engine quits, right, when the engine quits. So now we can look at the second phase and the second phase is going to be the when the engine fails. So engine bell face And the initial speed when the engine fails? Well, we know this is going to be the final speed from the vertical ascent phase. We talked about this. Okay. And we found that to be 390 m/s. Now the final speed, this is the final speed at maximum height. Okay, We're trying to figure out the maximum height and so we're gonna look at the final speed at the maximum height and remember that when we're looking at a maximum height, We have a speed of 0m per second. Okay? The space shuttle is going to momentarily come to rest as it goes from going upwards to coming back down and it'll be momentarily at rest right at its maximum height. And so we're looking at a final speed of zero. The acceleration is going to be the acceleration due to gravity negative 9.8 m/s squared. And we want to figure out Delta Y two. Okay. How high does the space shuttle reach during this engine fail phase or what is the vertical distance that the space shuttle covers during this phase. Now we're gonna use the equation the um equation without T. Okay, we have three knowns and the unknowns. We want to find this delta y two and so we have that V. F two squared is equal to V naught two squared plus two A two Delta Y two. Okay, so on the left hand side we just get zero on the right hand side we get 390 m per second squared plus two times negative 9.8 m per second squared times delta Y two. And if we saw for delta Y two, this is going to be equal to 390 m per second, all squared, divided by two times 9. m per second squared. And this gives us 7,760. m. And so the vertical distance that the space shuttle covers once the engine has failed is 7,760. m up to the maximum hype. Okay, now be careful when you're answering the problem. Okay, It's easy to go back and answer. Okay, we know the maximum height is 7760.2 m. But remember that this is the height or the distance vertical distance traveled. Once the engine fails before the engine failed, we had already gone 5070 m. And so the total height, the maximum height that this space shuttle actually reaches is the sum of these two values. And so the maximum height Is going to be Delta Y one Last Delta Y two, which is equal to 5070 m plus 7760.2 m for a total of 12, 0.2 m. Now, the answers you'll notice are in kilometers in order to convert to kilometers, we're going to divide by 1000 and when we divide by and take three significant digits, we're gonna have 12.8 kilometers, and that is going to be the maximum height that the space shuttle reaches. And so we go back to our answer choices and we found the maximum height attained by the space shuttle is going to be D 12.8 km. Thanks everyone for watching. I hope this video helped see you in the next one.

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s² for 30 s, then runs out of fuel. Ignore any air resistance effects. a. What is the rocket's maximum altitude?

Key Concepts

Newton's Second Law of Motion

Kinematic Equations

Conservation of Energy

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