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(II) A rocket rises vertically, from rest, with an accelerationof 3.2 $\mathrm{m} / \mathrm{s}^{2}$ until it runs out of fuel at an altitude of 950 $\mathrm{m}$ .After this point, its acceleration is that of gravity, down-ward. (a) What is the velocity of the rocket when it runs outof fuel? (b) How long does it take to reach this point?(c) What maximum altitude does the rocket reach? (d) Howmuch time (total) does it take to reach maximum altitude?(e) With what velocity does it strike the Earth? (f) Howlong (total) is it in the air?

a) $v_{f}=78 m / s$b) $t_{u p}=24 s$c) $y_{\text {total}}=1260 m$d) $t_{\text {total}}=32$ secondse) $-160 \mathrm{m} / \mathrm{s}$f) $t_{\text {total}}=48$ seconds

Physics 101 Mechanics

Chapter 2

Describing Motion: Kinematics in One Dimension

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Simon Fraser University

Hope College

University of Sheffield

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

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(II) A rocket rises vertic…

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(II) $(a)$ Determine a for…

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So we have this rocket that's accelerating upwards at 3.2 meters per second squared, and it reaches the height of 950 meters before it runs out of fuel. And there are a lot of parts. Um, but they're all pretty much just the same thing, just with different numbers. So the first part, we want to know its velocity when it reaches this height and we know eso we want to figure out the final velocity here. We know our initial velocity is zero. So when we use this equation, it simplifies pretty nicely. I mean, he was Delta. Why here? Because we're moving vertically in. Our acceleration is 3.2 meters per second. Squared his party. Um, so we're looking for our final velocity. This term is zero and look in 3.2 for acceleration, and delta Y is 950 meters because we're starting at zero, we're ending in 9 50 and B squared is 6080. So our final velocity is around 78 meters per second. It's pretty quick. Okay, for the second part now we want to know, uh, the time it takes to reach this height So it's just the same numbers using a different equation. This time we're going to use this one that has time in it. So that's why it's gonna be vizier T plus 1/2 itchy squared Delta wise 9 50 The zero is zero because we're starting at rest. Our acceleration is 3.2 and this gives us 9 50 ISS 1.6 he squares or cheese squared about 593.75 ish, and our final T is about 24 seconds doing a lot of rounding here, but it's okay. Uh, now part see, we want to know the maximum altitude it reaches. So from this point forward, we're gonna be going from after this point that runs out of fuel and is under freefall. And then if if any parts require total time or, uh, anything like that will just add in these values from the first parts we we know that we have 78 meters per 2nd 24 seconds. It's a part. See, we want to know the maximum altitude reaches and so we'll say that we know are we write down what we know. We know where initial velocity now is going to be 78 meters per second and our initial height. It's not gonna be 9 50 because we're starting from this this point. And so our delta y here we want to know what's Thea? That's what we want to know. Here, Delta y is gonna be why final minus y zero my initial, which is 9 50 So if we sell for Delta Y, then we can figure out our wife final our maximum altitude. It's just our delta y plus r Y zero. So if I use, uh, we want to find Delta y and we know our velocities one more thing, our final velocity is zero. Because once we reach maximum altitude, that's when our velocity zero this is gonna be our initial velocity squared plus two a delta y okay. And we know this is zero. Our initial velocity is 78. Our acceleration now I should say, is minus G because we're in free fall. So this is gonna be minus 9.8 and our delta wise what we're looking for and if you solve this, you get delta. Why is 310 meters and so our wife final are maximize Gonna be 3 10 uh, plus 9 50 which is 1260 meters. So that's our max height and part Dino. We want to know the time it takes to reach this altitude. So again, we'll go back to this equation. Uh, easier t plus 1/2 A t squared. And we know our delta. Why now is 310 RV zero was still 78. And we know all this information acceleration is minus 9.8. Now we have a quadratic, not a problem. Bring everything in one side. We get 4.9 t squared minus 70 80 plus 3 10 zero. And this gives s t equals about eight seconds. And this is the time that it takes to go from, uh, when the fuel runs out to the max altitude. Remember, we're always starting from that from this point. Ah, so our initial T will have to add that on. So it's eight plus 24 is 32 seconds is the total time it takes to reach the max altitude. Now, for Part E, we want to know the velocity that it has when it hits the Earth So again we'll go back to this equation. He squared. Is the initial squared Let's to a delta y And now our delta Y is gonna be minus 950 meters because we're going from height of 9 50 down to zero at the Earth and we want to know our final velocity. Our initial velocity here is 78 same as always, and acceleration is minus G. Delta. Why we know is minus 9 50 So RV squared comes out to be this and that means our velocity is 157 meters per second. And if we're counting direction to then it's negative. But you just want the speed eso It's 1 57 and f The last part is going to be total time in the air. Uh, so again, we'll use Go back to this equation. Don't know why it's easier t plus 1/2 a t squared. We're using this Delta y now. So minus 9 50 is 78 See, plus 1/2 minus 9.8. He squared. So we have a quadratic again. Ah, 4.9 t squared minus 78 T minus 9 50 equal zero. This is almost the same is this When we had a PPE here, but the delta y is different. That's just gonna change our time. And if you solve this quadratic, get T eagles 24 seconds and again, that's the time it takes from when it runs out of fuel to hit the ground. But it also took us 24 seconds to get to that height. 24 plus another 24 seconds. It's 48 seconds.

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