A Photon Star Drive The nearest star outside our solar system is about 4.3 y awa

Question

A Photon Star Drive The nearest star outside our solar system is about 4.3 y away. At one-tenth the speed of light, it would take 43 years to reach it. our present rocket technology cannot get us there. The velocity of a traditional rocket starting from rest is given by where V is the relative velocity of the gases (exhaust speed) that propel the rocket m is the initial mass of the rocket, and mois the mass of the payload. With highest thrust velocity we have presently 4000 m/s), reaching o.1c would require an initial mass that is e 00 arger than the payload. For even a 1 kg payload, the initial mass would have to be larger than the mass of the Universe. Ifwe could make the initial mass 10,000 times the mass of the payload, it would still take well over 30,000 years for the But what if we could use light, which would have an exhaust speed of c? Well, the equation above is not valid, but you now have the tools to figure it out. We state without proof, that the most efficient energy conversion possible would be a matter- antimatter engine that converts mass into light that is "ejected" like a rocket exhaust. Consider a rocket with payload mass mp and fuel mass mr. The fuel mass would be equal parts of matter and antimatter that could be brought together in a controlled annihilation reaction inside a perfectly reflecting chamber that directs the light (gamma rays actually) out the exhaust port. We start from rest as shown in the first diagram. After all the mass of the fuel has been expended, we have mp traveling speed v, or fractional speed B.

Explanation / Answer

a) According to the basic theory of conservation of energy and momentum if net force on the system is negligible then the momentum p=mv remains constant all the time.

again ignoring the internal and external forces the total energy in this case E= PC will remain constant

So at rest we have to consider two masses of rocket the mass mf which remain constant and propelant mass mp which changes with time

So the initial mass means Mi= mf + mp

where mp is the function of t=time

during final mass calculation due to burning we get only the mass which changes with time

Mf =mp

According to the theory of momentum conservation

(mf+mp)v' =mpv

and from the theory of energy conservation

P1C =P2C

or (mf+mp)v' *C =mpv*C

therefore considering ideal rocket equation we can write gamma= (mf+mp)^2 +mp^2/2mp(mf+mp) and also apply the fractional speed beta

b) From the ideal rocket velocity equation we know that deltaU = Veq ln(mf/mp)

now ideal delta u is the maximum velocity a rocket can use so we use delta U= C

using Veq= .1C and mp= 10^5kg we get

C=.1C ln (mf/10^5)

or 10 =ln (mf/10^5)

or (mf/10^5) = 22026.46

or mf = 22026.46 * 10^5 Kg

c) 4.3 ly= 4.3* 9.4*10^15m =S

total time given = 1year=3.15*10^7 seconds

using S=ut +1/2ft^2 we get f= 2S/t^2 = 2*4.3* 9.4*10^15/(3.15*10^7)^2 = 81.471m/s^2

using net force equation we get the fuel mass required.

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