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 \(3.84 \cdot 10^{21}\) 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?

Short answer

Question: Determine the speed of a spaceship after 1 hour, 1 week, and 1 month, when it is attached to a solar sail that gets propelled by reflecting photons. Also, find the time required to reach a speed of 8000 m/s. Solution: 1. Calculate the force exerted on the solar sail by the photons. 2. Calculate the acceleration of the spaceship. 3. Calculate the speed of the spaceship after various intervals. 4. Calculate the time it takes the spaceship to reach a certain speed (8000 m/s).

Step-by-step solution

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. \(F = N \times (2 \times 1.30 \cdot 10^{-27} \,\mathrm{kg \, m / s})\)

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.

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.

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: \(t = \frac{v}{a}\)

Key concepts

Photon Momentum

When we think of momentum, objects with mass often come to mind. However, even photons, which are particles of light with no mass, possess momentum. This might seem puzzling at first, but it can be explained by Einstein's theory of relativity.
Photons have energy and travel at the speed of light. According to the equation for photon momentum, \[ p = \frac{E}{c} \] where \( p \) is momentum, \( E \) is energy, and \( c \) is the speed of light. Because they have energy, photons exhibit momentum without requiring mass. This unique property enables them to transfer momentum upon interaction with objects, like a solar sail in space. Think of it as a gentle but constant push that can propel spacecraft over time.
If photons strike a solar sail and are reflected back, they essentially impart a change in momentum which we can harness for propulsion.

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This principle is crucial when considering the force exerted by photons on a solar sail.
When photons strike and reflect off the sail, they experience a momentum change. This results in an impulse applied to the photons. Importantly, the sail receives an equal and opposite impulse. As a result, the spaceship connected to the sail is pushed forward. This is the underlying mechanism behind the propulsion gained from a solar sail.
Understanding this law helps us visualize how seemingly small forces, like those from countless photons, can add up to significant propulsion over time.

Impulse and Momentum

Impulse is a fundamental concept when studying how objects interact. It is the product of the force applied and the time over which it is applied. Mathematically, impulse \( J \) is given by:\[ J = F \times t \] It is also equal to the change in momentum:\[ J = \Delta p \] For a solar sail, the force exerted by photons over time results in a continuous change in momentum of the attached spacecraft.
Each photon that strikes and reflects off the solar sail contributes a small impulse. With billions of photons striking the sail every second, the cumulative impulse can gradually increase the spacecraft's velocity.
This concept allows us to compute how quickly and efficiently spacecraft maneuver through the vastness of space.

Space Travel Calculations

Space travel, especially with solar sails, involves careful calculations. Key calculations stem from determining the force, acceleration, and duration for achieving various speeds.
For our 1000 kg spaceship, the force applied by photons is calculated using:\[ F = N \times (2 \times 1.30 \cdot 10^{-27} \, \mathrm{kg \, m / s}) \] where \( N \) is the number of photons striking the sail per second. Knowing this force, we find acceleration \( a \) using Newton's second law:\[ a = \frac{F}{m} \] Finally, velocity after a certain time \( t \) is determined using:\[ v = a \times t \] These equations help predict how the spaceship speeds up. Whether calculating how fast a ship travels after an hour or planning how long to reach a certain speed, precise calculations ensure successful navigation.
Being able to compute this means we harness the power of light for sustainable and efficient space travel.