Although they don’t have mass, photons—traveling at the speed of light—have mome
ID: 2064714 • Letter: A
Question
Although they don’t have mass, photons—traveling at the speed of light—have momentum. Space travel experts have thought of capitalizing on this fact by constructing solar sails—large sheets of material that would work by reflecting photons. Since the momentum of the photon would be reversed, an impulse would be exerted on it by the solar sail, and—by Newton’s Third Law—an impulse would also be exerted on the sail, providing a force. In space near the Earth, about 3.84·10^21 photons are incident per square meter per second. On average, the momentum of each photon is 1.30·10^–27 kg m/s. Assume that we have a 1287 kg spaceship starting from rest attached to a square sail 29.9 m wide.a) How fast could the ship be moving after 35 days?
b) How many months would it take the ship to attain a speed of 7.93 km/s, roughly the speed of the space shuttle in orbit?
Explanation / Answer
Part A)
The impulse momentum theorem states that Ft = p
The momentum of each photon is reversed, so the change in momentum is 1.37 X 10-27 - (-1.37 X 10-27) which is 2.74 X 10-27 kgm/s
The sail is 29.9 m wide, so its area is (29.9)(29.9) = 894.01 m2
Since 3.84 X 1021 photons are incident per square meter, that is (3.84 X 1021)(894.01) = 3.433 X 1024 photons
The force applied is then...
F = (3.433 X 1024)(2.74 X 10-27) = 9.41 X 10-3 N
From Newtons second law, F = ma
a = (9.41 X 10-3)/(1287) = 7.31 X 10-6 m/s2
35 days must be converted to seconds. (24 hrs/day and 3600 sec per hour) = 3024000 sec
Using vf = vo + at we can find the velocity
vf = (0) + (7.31 X 10-6)(302400) = 2.21 m/s
Part B)
To get to 7930 m/s,...
7930 = (0) + (7.31 X 10-6)(t)
t = 1.08 X 109 sec
That many seconds is 3.01 X 105 hrs, or 1.26 X 104 days, which is 34.4 years
34.4 years is 412.8 months
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