This is problem 49 in the book. For 30 seconds this rocket is accelerating from Earth at the rate of 30 meters per second squared. We're not taking account air resistance. We have two questions we're going to find out the answer to. What is the maximum altitude and how long is it in the air? To solve this problem I'm going to break this up into three phases. But we have the phase one which lasts for 30 seconds is given to us. That's power flight. We are accelerating upwards. We also have this growing velocity in the upwards direction. And after 30 seconds we enter phase two. That is we run out of fuel. We're no longer accelerating upward. Instead we're decelerating towards Earth at 9.81 meters per second squared. Earth's gravitational field. However, we're still accelerating or we're still have this this velocity taking us higher to higher altitude. And we're going to still continue increasing an enormous amount of altitude during this phase two. In phase three begins of when we have a helicopter, when we have a zero-pilot in the middle of a three-row, we're still have this velocity. This velocity is not going to be taking us higher to higher altitude. And we're going to still continue increasing an enormous amount of altitude during this phase two. Phase three begins of when we have a zero velocity and we enter a free fall return back to Earth. And a cell of these three times is how long it's in the air. We can find out the maximum altitude at the end of our phase two of calculation. And for solving this you can actually combine phase two and phase three. But I think it's easier to break them up and do them all individually. And so we'll use any equation from our book where our distance formula that's x equals x zero, our initial but the offset which is zero in this phase because we're at ground level. Our initial velocity times t, initial velocity is zero because we start at rest. Plus acceleration divided by two times t squared. And all these units end up being after more implications are in units of distance. Meaders per second, time second is meters. Meaders per second squared times second squared is also meters. So these will all add up. As we're on the ground and we're taking off at a acceleration of 30 meters per second. We're going to do this for 30 seconds. So how high do we go? What's our offset here is zero. Our initial velocity is zero. What we still have here is 30 over two for 30 second squared. And so our x is equal to 15 times 30 squared is 900. This altitude is 13,500 meters. And that's our elevation. We're at when we have run out of fuel. We are at the end of our phase one. This is our phase one altitude. Our phase one time is 30 seconds. It was given to us in the problem. We're going to begin our phase two. We're still moving upwards at a positive velocity. But we're being decelerated back down to earth. So we see here at beginning of phase two, our velocity is 900 meters per second. Very, very fast. It's a kilometer or second. Now I know how long does it take? How long are you in phase two? Well, we're decelerating by 9.81 meters per second squared. So 9.81. We do this division meters per second square. When we divide this, our units, our meters, and most of our seconds will cancel out. We'll left with the unit of seconds. 995 by 9.81 is actually 91.7 seconds. And we're going to use this time to find our distance. We traveled over this and out of time. x equals x is 0 plus v is 0. T plus a over 2 times t squared. This would be the most sophisticated equation in the whole problem. We have our initial offset here. We're at an altitude of 13,000 meters, 13,500 plus a of v is 0. Our v is 0 is 900. So we're beginning of phase two. We're at 900 meters times times. And our acceleration, this time our acceleration is a negative number because we're decelerating back down towards Earth. That's 9.81 divided by 2. 9.81 divided by 2, t squared. And that's our new altitude. When we take our t that we solved earlier, that's say 91.7 seconds. And substitute it in this equation, we will find our total altitude that we have climbed to all the way up until we've run out of velocity. That is our maximum altitude to answer to part a. What we have is 13,500 plus if you multiply 900 times 91.7, you'll get 82,568. We did not pay attention to all these significant digits. It's not really that precise. We're going to trim this part of the. And our altitude we lose due to this negative 9.81. The accelerating of the downwards is going to be a negative 41,284. This was supposed to be a negative number. And so what we find that this is equal to is that is 54,784. That's what it says on the calculator. Correct significant digits is 54.8 kilometers. That's our maximum altitude, 54.8. And so at this point here, we have a velocity of zero. We are 54.8 kilometers up. And now we are beginning our phase three, our freefall back to earth. And all we need to know now is how long does that take? Because to put overtime into air is to free fall time plus this 91.7 plus the 30 seconds it took for us to burn up that fuel. And to some of those three times, that's what we're looking for. And so we take our distance equation. There x equals x zero plus v zero times time plus that a over 2t squared. That a once again is equal to 9.81. And our x zero is our 54.8. I'm going to write it in full precision here. 54784 minus 9.81 over 2t squared. Our resulting altitude at the end of this is going to be zero. So our final condition here, our final offset, the altitude is going to be zero. So what we do is we solve this for t. So we have 9.81 over 2t squared equals 54784 multiplied both sides by 2. And then divided by 8.1 we get t squared equals 1169. Square root of the little sides here we get t is equal to 105.7 seconds. And so what's our time in flight? It's the sum of this phase three plus the time for phase two, 91.7 plus 30. We have 0.4, 677.4, 1227.4 seconds. And is our