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Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket's mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of $250 \mathrm{km},$ and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of $1.00 \times 10^{5} \mathrm{kg}$ (100 tons), what total yield nuclear explosion in tons of TNT is needed?

(a) $\frac{\Delta m}{m}=2.73 \times 10^{-14}$(b) $\Delta m=2.73 \times 10^{-8} \mathrm{kg}$

Physics 103

Chapter 28

Special Relativity

Relativity

Cornell University

Hope College

University of Winnipeg

McMaster University

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Well, the total rest mass energy of an object a vague most m is given by the knot is equal to Connecticut Energy, Less potential energy. Andi here m is, uh mm. Is muscle the material months off? The material material on dhe see is the speed of light. Ah, speed of light and C equals 3.0 Multiply by 10 to the power eat. Made her for a second. Well, we know that kind of energy is given by get the G dimes. M time small m divided by you are square well here. Jeez, the gravitational constant. And is, uh, uh, the muscle here and small m is muscle off the object, and capital R is the distance off the object from the center of the earth. So are is the distance. Are is the distance off Uh, the object from the same dear Oh, dear. From the center off the earth, off the art All right. And potential energy is given by MG edge. We have ah, height off the rocket X equals, uh, 250 kilometers. And 250 kilometers. It's equal to 2 50 Multiply by 10 to the power of three meters and we have radius of the earth. Small art is the radius off the earth. Keeps 6.37 Multiply by 10 to the power six meters. All right and ah, distance off the rocket from the center of the earth is capital R, and it's equal to our less edge. And this gives us, uh, small R is explained 37 multiplied by 10 to the power six less edge is 2 50 Multiply by 10 to the power of three and therefore are equal 6.62 Multiply by 10 to the power six meters. Well, no, let's calculate till Connecticut Hershey on kinetic energy is, uh, captain Jew capital M small M, divided by two are square. Let's plug in the values capital do 6.67 multiplied by 10 to the power minus 11. Multiply by and capital M uh, months off the earth is 5.972 Multiply by 10 to the power 24 kilograms multiply by mars of object divided by to multiply They are is 6.62 multiplied by 10 to the power six whole square and then four kinetic and Rashid is equal to four point 55 uh, 4.55 Um, mater made her square, uh, four second square and potential energy is equal to m t H G's mine pointed multiplied by a cheese to 50. Multiply by 10 to the power three meters and therefore potential energy equals 2.45 multiply by 10 to the power tree made her made her square or a second squared and the total energy e not is the sum of kinetic and potential uh, the second attic energies 4.55 less on potential energies to 0.45 Multiply by 10 to the power three and this gives us 2.46 2.46 multiplied by tentative over three meter, meter, square or second square and we know they dealt M is equal to the non divided by C square. Don't I am is equal to me. Not divided by C square, we have we not Well, you know what is, uh, 2.46 Multiply by 10 to the power free divided by C square. Well sees 3.0 multiplied by 10 to the power it holds square and dividing the two numbers. We have 2.73 Multiply by identity baller. Um, stand to the power minus minus for Dean kilogram. And therefore, uh, uh, Delta em, divided by m, is equal to dew 0.73 Multiply by 10 to the dollar minus 14. And according to the question, if M is equal to 1.0 multiplied by 10 to the power five kilogram, five kilogram, then, uh, Delta M then don't m is equal to 2.73 Multiply by 10 to the power minus for Dean, don't jammies minus 14. Um, multiply by, um, damn dirty power. Five kilogram and therefore don't A M is equal to two point 73 Multiply by 10 to the power minus eight kilogram.

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