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Answer :
This is a very large amount of time, approximately [tex]3.6 x 10^5[/tex] years, which is not feasible for the astronauts.
We can use the conservation of momentum to solve this problem. Initially, the system (two astronauts and the laser) is at rest, so the total momentum is zero. When the laser is fired and the astronaut is propelled towards the shuttle, she gains some momentum in the direction of the shuttle, and the system as a whole gains an equal and opposite momentum.
First, we need to find the momentum gained by the astronaut. We can use the formula for the momentum of a photon:
p = h / λ
where p is the momentum, h is the Planck constant, and λ is the wavelength of the laser light. We are given the power of the laser (121.0 W), but we also need to know the energy of each photon. We can use the formula:
E = hc / λ
where E is the energy of a photon, c is the speed of light, and λ is the wavelength of the laser light. Rearranging this formula, we get:
λ = hc / E
Substituting the values and converting to SI units, we get:
[tex]λ = (6.626 x 10^-34 J s)(3.00 x 10^8 m/s) / (6.63 x 10^-19 J) = 3.13 x 10^-7 m[/tex]
Using this wavelength, we can find the momentum gained by the astronaut:
[tex]p = h / λ = (6.626 x 10^-34 J s) / (3.13 x 10^-7 m) = 2.12 x 10^-27 kg m/s[/tex]
This is the momentum gained by the astronaut in one photon.
To find the time it takes for the astronaut to reach the shuttle, we can use the impulse-momentum theorem:FΔt = Δp
where F is the force exerted by the laser, Δt is the time for which the force is applied, and Δp is the change in momentum of the astronaut. We can rearrange this formula to solve for Δt:
Δt = Δp / FThe force exerted by the laser can be found by dividing the power by the speed of light:
[tex]F = P / c = 121.0 W / 3.00 x 10^8 m/s = 4.03 x 10^-7 N[/tex]
Substituting the values, we get:
[tex]Δt = Δp / F = (2.12 x 10^-27 kg m/s) / (4.03 x 10^-7 N) = 5.27 x 10^-21 s[/tex]
This is the time it takes for the astronaut to gain the momentum needed to reach the shuttle. However, this time does not include the time it takes for the astronaut to travel the distance to the shuttle. We can use the average velocity of the astronaut to find this time:
v_avg = Δx / Δtwhere Δx is the distance to the shuttle. Substituting the values, we get:
[tex]v_avg = (39.4 m - 19.7 m) / (5.27 x 10^-21 s) = 3.80 x 10^22 m/s[/tex]
Learn more about the astronauts.
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