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A model rocket blasts off and moves upward with an acceleration of 12 $\mathrm{m} / \mathrm{s}^{2}$ until it reaches a height of $29 \mathrm{m},$ at which point itsengine shuts off and it continues its flight in free fall. (a) What isthe maximum height attained by the rocket? (b) What is the speedof the rocket just before it hits the ground? ( c) What is the totalduration of the rocket's flight?

(a) $g=-9.81$(b) $35.57 \mathrm{~m} / \mathrm{A}$(c) $8.52 \Delta$

Physics 101 Mechanics

Chapter 2

One-Dimensional Kinematics

Motion Along a Straight Line

Rutgers, The State University of New Jersey

Simon Fraser University

McMaster University

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Okay, So in this problem we have a rocket moving upwards. If our acceleration off to Ralph meters per second square, the rockets maintain his deceleration until he reaches the height off 29 meters. After this moment after disposition Hey, shut off his changing and continue its flight in free fall. Okay, so the first question is, what is the maximum height? Let's go Ex Toto off the after rocket What is the maximum height that the rocket reaches? Okay, so first of all, let's think about entire movement. Let's see, we have our rocket moving from rest with acceleration of 12 meters per second squared. Then the rockets reaches first position. Ex ICO swan disposition. He shut offs his angel and continues moving upwards. And so he reaches a maximum height school ex too, and stop! The thing that we have to notice is that in this segment off the movement, his acceleration is the gravity's acceleration off miners nine point 81. So actually we have two movements. The first movement is ah accelerated movement. And the 2nd 1 is a decelerated movement. So we need to use two equations to calculate it, each off them separately. Okay, so let's calculate this. The first part of the movement is the accelerated move movement. So we want to find out the height. So let's see, to calculate the maximum height, we need to discover the velocity. So, in the first part of the movements, we have these where it goes easier square. Well, us too. A doubt ex. Sorry. That's call X one. So you guys can not get confused. This is going to be ex. What? Okay, so V, we don't know. The zero is zero because he starts addressed, plus two times 12. And the first position we know is 29. So actually, the first velocity, the maximum velocity, it's going to be 69 six. So the square root 696 It's just 26.4 meters per second. So this is the first velocity. This is the velocity. A position X one. When we have the position X one, the final velocity, his 26 point for. Okay, so come back here. Now we have the velocity off the final position. We can just cover there. Position x two. So the position next to we're going to use this immigration is just going to be the square equals to visit or square. Well, it's true. G Delta Not doubt. That's put ah X two in here x two. So let's see. Their final velocity is your toe. The initial velocity is the fine of a lost such of the first part. So it's just 69 Six miners, two times nine points. 81. It is the gravity's acceleration now pointing downwards times the x two. Let's get these X here. Okay, so let's see the distance x two Just going to be, uh let's see. 35 35 point 47 meters. So the answer to the first item, What is the maximum height that we call ext E X Toto? Just going to be x one plus x two. So this is finally 64 point 47 meters. This is the maximum height of the rockets. Okay, Next item the next item we need to calculate what is the speed of the rocket just before reaching the ground. Okay, so this time now we have 1/3 part of the movement. Let's come back in here. See? Still rockets stop. Here we begin a turd art of the movement that is, the rockets beginning to fall under the gravity's acceleration one to reach the ground. This is a free falling object. So in this part, off the movement, we can use again the same equation which are me to square because the zero less too times g Delta X So v zero is what we want to discover is your RV is what we want to discover Vizier is, you know, because in the beginning of the third part of the movement the beginning of the dirt art off their movement, the rocket starts from rest. So these ears just sooner. Plus two times nine point 81 Delta X Delta access the Toto space. The maximum height. It was just going to be 64 0.47. So calculating this, we have the answer to this second item. The velocity of the rocket just before reaching the ground is going to be dirty. Five point 57 meters per seconds. Okay, so the dirt item the eye can see we want to discover. I did told told oration off the rockets flights. So the total duration of the rockets fight we just need to use what information we have from the previous calculation. So we just need to use the vehicles, visit Europe, Loss 80. So let's see. First of all, we need to calculate the time off each off their movements separately because you don't have to. We don't have a way to calculate it all the time. In one single calculation, we need to separate this movement. So the first part of the movement we start with initial velocity of zero with acceleration of 12. We need to find a time in the final velocity we know, which is. It's just 96.4. So this is going to be 26.4. So their first time going to be 26.4 divided by 12. This is a code 2.2 seconds. The second part off their movement. We have another time. Vehicles 30 plus GT. This time we're going to have V zero. V zero is 26.4 minors, 9.81 e. So the second time it's going to be 26.4, divided by 9.81. This is just 2.7 seconds and the final part of the movement. They're free. Falling from his maximum right height is just vehicles Visitor loss GT so V we already calculated this was 35 0.57 s is equal to zero plus 9.81 key. So the final time waas 35 point 57 divided by nine point 81. This is going to be equal to three points 62 seconds. So the answer to the final letter final writing is that total time it's going to be to a point too. Plus 2.7 lost 3.62. This is going to be equal to eight point 52 seconds and this is the final answer to the problem. Thanks.

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