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(1) Jill and her friends have built a small rocket that soonafter lift-off reaches a speed of 850 $\mathrm{m} / \mathrm{s} .$ How high abovethe Earth can it rise? Ignore air friction.

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$3.7 \times 0^{4} m$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

University of Washington

University of Sheffield

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Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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(I) Jill and her friends h…

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all right for this problem, we're gonna use the definition of gravitational potential energy, which is, uh, you equals negative ci times m times Small lamb over our, uh are being the distance between the two objects and between the two centers of mass and being the two messes. Ah, And so here we equate ah, energy at the ground. So that would be kinetic energy at blessed Ah, take off of the rocket. It's called a case of our plus potential energy of the, uh, rocket at the surface of the earth. So are will be radius of the earth on that will equal Ah, assuming no affection at all equal potential energy at a height which above the surface called the use of each. So this is 1/2 m v, not square. Do not being the velocity of the launch velocity of the rocket plus negative because that's new. Negative, negative. Ci times Bigham is a massive beer. It small m is the mass of the rocket. Same is this up over our so be it. That being the radius off the earth, that is a quote too Another again, a negative g m E times m over this time it's raising their two plus the height above above the ground. So the rocket reaches okay on. So we notice immediately the small arms cancel, which is good. We don't need to worry about the massive will rocket. She's not given anywhere. Ah, and so we have one. Have you not square people cheer me up. Very like, uh, minus Jeremy Ever are equals negative. P m E over Ari plus h. Okay, so we want h. And so for that we need to do a little algebra here. Um, we have 1/2 view, not squared. Uh, equals g m me as common taking not as common And pulling this term out to the right hand side. So becomes a negative to ah, so the it's most fly by one of our party and subtract minus one over our subtract one of our e plus inch, uh, to that one. And this is what you have. Okay, um, then do one other thing. Now, uh, so one of radius of the earth. What's one of radius of the earth? Uh, minus one of her race of the earth. Plus height will be equal to I'm just pulling out the G m e uh, to the other side will be quote to be not squared over two She m e. Okay, um and then you have Then you have one of her r e plus h. He calls that term negative view, not squared. Uh, are, um, looks like right. So you have negatively not squared over to G m e Um, plus one over r e. And the reason this term becomes negatives that we're just pulling one of our e plus age to the right hand side to the other side again and be not squared over to Jamie to the left hand side. Great. With another one of our reach. Okay. And so this looks like a you know, not an easy task. Thio, uh, pull out just h. And so what we're gonna do is we're gonna inverse both sides, and what we have is our e h is equal to one of our e finest. Do not squared over to G m e in verse. Andi, therefore h just age the height is all of this stuff one of ari minus B not squared over to g m u in verse. Uh, minus party. Okay. And there are other algebraic simplifications you could do. But this will honestly do all you have to do. At this point, it's plug in values and be careful when you're using the calculator. And so what you have is one of our e Sorry radius of the earth is about 6330 kilometers. So 6300 well 68 0 kilometers. Ah, Therefore, it's six bad times 1000 meters. So that's 6.3 times 10 to the six meters. Okay, six meters minus V, not squared rate velocity of launch. That's 8 50 meters per second square. Um, over two times Jew, which is six point 67 times 10 to the negative 11 um, in s i u nit. You need to worry about the units If you use that, um times massive dear it, which is about 5.98 times 10 to the 24 kilograms. By the way, this information is all in the book. Uh, so are you can look it up. Everyone knows the massive made use of the earth. So no, you know these quantities. And so in verse, that and you subtract radius of the earth, which is again 6.38 tons tend to the six meters. So very careful when you're plugging these into the calculator on, then and so what you end up getting is that, um the height, the maximum height to which it can rise is 3.708 times 10 to the four meters or, uh or 3708 kilometers. And and I think that is it and you look like that, is it?

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