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 \(3.84 \cdot 10^{21}\) photons are incident per square meter per second. On average, the momentum of each photon is \(1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). For a \(1000 .-\mathrm{kg}\) spaceship starting from rest and attached to a square sail \(20.0 \mathrm{~m}\) wide, how fast could the ship be moving after 1 hour? One week? One month? How long would it take the ship to attain a speed of \(8000 . \mathrm{m} / \mathrm{s}\), roughly the speed of the space shuttle in orbit?

Step by step solution

01

Calculate the force exerted on the solar sail by the photons

Since the sail reflects the photons, the momentum of the photons is reversed. Therefore, the change in momentum of a single photon would be \(2 \times 1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). Let's calculate the total force exerted on the solar sail due to all the photons incident on it. The area of the square sail is \(A=(20.0)^2 \,\mathrm{m^2}\), so the total number of photons incident on the solar sail each second is \(N = A \times 3.84 \cdot 10^{21} \,\mathrm{photons/ m^2/s}\). Now, let's find the total impulse exerted by all the photons per second, which is equal to the total force exerted on the solar sail. \(F = N \times (2 \times 1.30 \cdot 10^{-27} \,\mathrm{kg \, m / s})\)

02

Calculate the acceleration of the spaceship

Now that we have the force exerted on the solar sail, we can find the acceleration of the spaceship using Newton's second law: \(a = \frac{F}{m}\) where \(m=1000 \, \mathrm{kg}\) is the mass of the spaceship.

03

Calculate the speed of the spaceship after various intervals

Now that we have the acceleration, we can find the speeds at different time intervals using the equation of motion: \(v = a \times t\) We will calculate the speed of the spaceship after 1 hour, 1 week, and 1 month.

04

Calculate the time it takes the spaceship to reach a certain speed

We are also asked to find how long it would take the spaceship to attain a speed of \(8000\, \mathrm{m/s}\). To find this time, we will use the same equation of motion: \(t = \frac{v}{a}\)

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