You are the sole crew member of the interplanetary spaceship...

You are the sole crew member of the interplanetary spaceship $T : 1339$ Vorga, which makes regular cargo runs between the earth and the mining colonies in the asteroid belt. You are working outside the ship one day while at a distance of 2.0 AU from the sun. II AU (astronomical unit) is the average distance from the earth to the sun, $149,600,000 \mathrm{km} .$ I Unfortunately, you lose contact with the ship's hull and begin to drift away into space. You use your spacesuit's rockets to try to push yourself back toward the ship, but they run out of fuel and stop working before you can return to the ship. You find yourself in an awkward position, floating 16.0 $\mathrm{m}$ from the spaceship with zero velocity relative to it. Fortunately, you are carrying a $200-\mathrm{W}$ flashlight. You turn on the flashlight and use its beam as a "light rocket" to push yourself back toward the ship. (a) If you, your spacesuit, and the flashlight have a combined mass of 150 $\mathrm{kg}$ , how long will it take you to get back to the ship? (b) Is there another way you could use the flashlight to accomplish the same job of returning you to the ship?

Key Concepts

Alternative Photon Propulsion Methods

Beyond directly using the momentum of emitted photons, another approach is to use reflective techniques. For instance, a light beam could be directed toward a reflective surface, causing the photons to bounce off and thereby transferring momentum. This alternative method can be used to either accelerate oneself or to manipulate another object's motion, depending on the configuration of the reflective surfaces and the direction of the light beam.

Thrust from Light Emission

By relating the power output of a light source to the force produced, one can deduce that the thrust (force) is equal to the power divided by the speed of light. This conversion is central to estimating the acceleration obtained by an object when it uses its own light emissions for propulsion. It links the energy characteristics of the light source to the mechanical motion of the object.

Photon Momentum

Despite having no rest mass, photons possess momentum proportional to their energy divided by the speed of light. This nonzero momentum allows light to exert a force when it is emitted or absorbed. Recognizing the relationship between light energy and momentum is crucial in understanding how a beam of light can produce thrust.

Radiation Pressure

Radiation pressure arises because photons, when they strike a surface, impart momentum to it. This pressure is the source of the force that can be exerted by light and is harnessed in applications such as solar sails. In propulsion, the cumulative effect of many photons can generate a measurable force even if each individual photon's momentum is very small.

Conservation of Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. In the context of using light for propulsion, the momentum carried away by the light must be balanced by the momentum gained by the emitting body. This is the foundational idea behind getting a reaction force from emitting photons, even though the photons are massless.

Transcript

0:00 Hello everyone.

00:01 This is a really fun problem in which we are asked to find out how long it's going to take us to get back to our ship near an asteroid belt, given that we can only use a flashlight as a rocket, so to say, quote unquote, rocket, to propel ourselves back to the ship.

00:20 So we're told that our spacesuit and every equipment that we have on us is weighing 150 kilograms.

00:30 And we're told that the flashlight has a power output of 200 watts.

00:36 And we are a distance of 60 meters away from the spaceship.

00:41 And we have no relative velocity with respect to the ship, meaning that our initial velocity towards the ship is zero meters per second.

00:51 So this is a really fun problem that might come useful in like 100 years if you are ever stock outside of your spaceship.

01:00 So what do we do? so first we have to find the momentum flow rate of the electromagnetic radiation that's coming out of the lamp.

01:11 So that's what these red errors here represent.

01:14 And so that's given by one over the area over which the power.

01:21 Is being put out.

01:23 So that's going to be like, you know, the roughly the face of the lamp over which the lamp shines or the light shines.

01:32 And this is multiplied by the time derivative of the momentum transfer or multiplied by the momentum transfer, which is the time derivative of the momentum.

01:43 Okay.

01:43 And so this is given by the pointing or the magnitude of the pointing vector divided by the speed of flight, which is equivalent to saying pretty much that, you know, if you're time averaging the pointy vector, then you get that this is the intensity divided by the speed of flight.

02:00 And then using the fact that the intensity is the power over the area of which the power is put out as well.

02:07 So this area is again the lamp's face.

02:12 So given that the intensity is given by this, we see that this quantity over here...

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