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.-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? After 1 week? After 1 month? How long would it take the ship to attain a speed of \(8000 . \mathrm{m} / \mathrm{s}\), roughly the speed of a space shuttle in orbit?

Short answer

Question: Calculate the speed of the spaceship with a solar sail at 1 hour, 1 week, and 1 month, and determine how long it takes to reach a speed of 8000 m/s. Answer: The speed of the spaceship after 1 hour is 14.36 m/s, after 1 week is 2415.20 m/s, and after 1 month is 10340.80 m/s. It will take approximately 23 days, 2 hours, and 10 minutes to reach a speed of 8000 m/s.

Step-by-step solution

Calculate the Force exerted by photons

We are given that \(3.84 \cdot 10^{21}\) photons are incident on each square meter per second and each photon has a momentum of \(1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). Since a photon's momentum is reversed on reflection, the change in momentum per photon is \(2 \times 1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). To find the total change in momentum per square meter per second due to all the photons, we multiply the change in momentum per photon by the number of photons: \(\Delta P = 2 \times (1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}) \times (3.84 \cdot 10^{21} \mathrm{~photons}/\mathrm{s}) = 9.98 \cdot 10^{-6} \mathrm{~N}\) The solar sail has an area of \(20.0 \mathrm{~m} \times 20.0 \mathrm{~m} = 400.0 \mathrm{~m^2}\). The total force exerted on the solar sail is: \(F = \Delta P \times \text{Area} = (9.98 \cdot 10^{-6} \mathrm{~N})\times (400.0 \mathrm{~m^2}) = 3.99 \mathrm{~N}\)

Find the acceleration of the spaceship

Using Newton's second law (\(F = ma\)), we can find the acceleration of the spaceship with a mass of \(1000\) kg: \(a = \frac{F}{m} = \frac{3.99 \mathrm{~N}}{1000 \mathrm{~kg}} = 3.99 \times 10^{-3} \mathrm{~m/s^2}\)

Calculate the speed after 1 hour, 1 week, and 1 month

Using the formula \(v = at\) where a is the acceleration and t is the time, we can find the final speed after 1 hour, 1 week, and 1 month: 1 hour: \(t = 3600 \mathrm{s}\), \(v = (3.99 \times 10^{-3} \mathrm{~m/s^2})(3600 \mathrm{s}) = 14.36 \mathrm{~m/s}\) 1 week: \(t = 7 \times 24 \times 3600 \mathrm{s} = 604800 \mathrm{s}\), \(v = (3.99 \times 10^{-3} \mathrm{~m/s^2})(604800 \mathrm{s}) = 2415.20 \mathrm{~m/s}\) 1 month (30 days): \(t = 30 \times 24 \times 3600 \mathrm{s} = 2592000 \mathrm{s}\), \(v = (3.99 \times 10^{-3} \mathrm{~m/s^2})(2592000 \mathrm{s}) = 10340.80 \mathrm{~m/s}\)

Calculate the time to reach 8000 m/s

To find the time it would take to reach a speed of \(8000 \mathrm{~m/s}\), we can use the formula \(t = \frac{v}{a}\): \(t = \frac{8000 \mathrm{~m/s}}{3.99 \times 10^{-3} \mathrm{~m/s^2}} = 2005025 \mathrm{s}\) 2005025 seconds is approximately 23 days, 2 hours, and 10 minutes.

Key concepts

Momentum of Photons

One might ponder how light, which seems weightless, could exert any pressure or force on an object. The fascinating answer lies in the enigmatic world of quantum mechanics, where particles of light, known as photons, do indeed possess momentum despite having no mass. According to the principles of physics, momentum is the product of mass and velocity. However, photons, while massless, derive their momentum from their energy and the constant speed of light, denoted by the equation \( p = \frac{E}{c} \), where \( p \) is momentum, \( E \) is energy, and \( c \) is the speed of light. This concept is critical in understanding the mechanics of solar sails, as the momentum transfer from photons to the sail propels the spacecraft through space.

When photons strike a surface like a solar sail, they reflect off, and due to the law of conservation of momentum, they impart a push onto the sail. This push is minuscule for each photon, but across the vast sea of light particles hitting a large sail, the cumulative effect can propel a spacecraft. The step-by-step solution for the solar sail problem involves calculating this momentum transfer by considering the change in a photon's momentum upon reflection, given as twice its original momentum, because it effectively reverses direction upon reflection.

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This fundamental principle applies universally, from the motion of celestial bodies to the operation of spacecraft. It becomes particularly interesting when contemplating the mechanics of a solar sail. As photons collide with the sail and change direction, they exert a force on the sail (the action). By Newton's Third Law, the sail exerts an equal and opposite force on the photons (the reaction).

This exchange can be puzzling because it involves light—a form of energy—interacting with a physical object, leading to a change in momentum for the spacecraft. The exercise solution applies this law to determine the impulse exerted on the photons and subsequently on the sail, which translates into force. By harnessing the continuous force exerted by sunlight, a solar sail-equipped spacecraft can gradually build up speed over time without the need for propellant, which is a significant advantage in space travel.

Spacecraft Acceleration

Acceleration is the rate of change of velocity of an object. In space, solar sail technology can be used to accelerate a spacecraft without the use of fuel. This is achieved through the constant pressure exerted by sunlight on the sail. The exercise solution calculates the acceleration of the spacecraft using Newton's second law, \( F = ma \) where \( F \) is the total force applied by the reflected photons, \( m \) is the mass of the spaceship, and \( a \) is the acceleration.

The calculated acceleration is constant, as the force exerted by photons is continuous and steady over time in the near vacuum of space. There's no air resistance or other forces to contend with, as would be the case within Earth's atmosphere or on its surface. This allows the spacecraft to continue accelerating as long as the solar sail is deployed and facing the sun. The resulting speeds after specific periods are determined by multiplying the acceleration by the time intervals, showcasing the potential for solar sails to reach impressive speeds over time, making them a promising technology for future long-distance space exploration.