Problem 3 In the future, we may use miniaturization to explore our has been sugg

Question

Problem 3 In the future, we may use miniaturization to explore our has been suggested as an efficent mode of travel. Let's consider a variant where a very su which a mirror attached and is accelerated by precisely aimed laser impart a momentum kick of Ap. We want to explore this possibility in differen approximations tomake The concept of a light sail small scientific payload pulses. These pulses reflect off the mirror and t ways. Here are the details and local region of the galaxy. The scientific payload and perfect mirror have a mass of m-1 kg Each pulse of light can be considered like a single particle (eg. photon ) and reflects directly back on itself. The rate of pulse emission is 10 Hz, as observed in the Earth-fixed frame First consider a low-velocity scenario where the payload's momentum can be cousi will get us a rough idea of what sort of laser source might be needed. Suggestion: It is classical monientum as pe-meaß and later the relativistic monsentum as ßme.I This easier to express the the payload's moment a) How much of a momentum kick, Ap, does the payload receive during each reflectionI we can consider the momentum of the pulse to have the same magnitude when it r eflects off of the mirror (ie. ach energy does each pube Dod to have in order to get the payload up to 20% the speed we are not considering conservation of energy, just momentum). Question: How of light, -0.2, in l year (i.e. 3 x 107 s). Also state your answer as a power (in watts). Next, let's consider the relativistle scenario. To solve it as before would take too long for the time available, thus we'll make a simplification that will make it more approachable. b) Now figure the relativistic result to get the payload up to -y Corsider it as a single interaction of light with the payload initially at rest. Determine the total energy of light required Eo considering that it reflects back with nonnentuin (i.e. energy) EO. e) From your answer in b) and assume our light source has power as calculated in part a), determine how long (in years) it would take to get the payloed up to this speed. ß-aLoreatz transform the photon's 4-1n menatum q-G d) When the pay payload's frame (where B, is the pulse energy found in part a)). What is the energy of each incoming pulse in this frame? (Consider whether the energy should be less than, greater than, or the same as E) to the

Explanation / Answer

d) one of the Einestein postulate of special thoery of relativity

speed of light remains same in all frames of reference.

Photon is a mass less particle, hence

momentum of photon in any frame of reference is E/c

or cp = Ey

Four mentum of photon in any frame is ( Er , Er )

hence the enrgy of each incoming pulse is same as Er

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