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II) $(a)$ Determine the rate at which the escape velocity from the Earth changes with distance from the center of the Earth, $d v_{\text { esc }} / d r,$ (b) Use the approximation $\Delta v \approx(d v / d r) \Delta r$ to determine the escape velocity for a spacecraft orbiting the Earth at a height of 320 $\mathrm{km} .$

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A) $-\sqrt{\frac{G M_{\mathrm{E}}}{2 r^{3}}}$B) $1.09 \times 10^{4} \mathrm{m} / \mathrm{s}$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

University of Washington

Hope College

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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we need to find out the rate of change off this cave velocity from the how it changes with distance from center of the now the escape velocity formula is actually given by the escape is square root off too G mass off earth divided by our Now, this can actually be written as are to the part off, excavated and as our to the par of minus half out big. This are in the new major to GM. Hey, now you find out the rate off change or the D V escape velocity with respect to the radius, Howard is changing. So we actually finding the generative of this velocity with respect to our so we'll find the derivative of this, which will be minus half are to the par off minus one by two minus one times do G m e. So that should give you minus have are to the part off minus three by two to g m e. So just simply applying the by the formula n xn minus one. But simple, dear innovative here. So if you sold that, simplify that he will get square root off G emi over too far cubed so that has the rate of change off the velocity with respect to or now they're saying that we need to use an approximation that the delta we escape is equals. Two d v Escape, divided by D are so the innovative of velocity with respect to Arthur Times Delta are we have already guard this d by D. R v escape, which is negative. G m e by two are cute times Delta, Are we just looking in? The rally was here, so you will get 6.67 times turned upon negative 11 times the mass richest 5.98 times turned to the par off 24 divided by two times 6.38 times 10 to the part or six cubed. So if you solve that, then he and then sorry. This is also more deployed by delta hour which is 3.2 times turned to the par off five meters which was 320 kilometers. So we're supposed to find out the escape velocity at a height of 3 20 kilometers used this approximation So if you sold that you know holding in the calculator you'll get negative to 80 meters per second. So this actually means that the escape velocity has decreased by 20 meters per second at this height. So therefore we'll have the velocity of escape is 1.12 dimes turning to the bottle for minus to 80. And if you saw that he will get approximately one point or nine times turned to the far off for meters per second. So that is the escape velocity, actually, at a height off CNN 20 kilometers above the surface, off the art.

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