When antimatter (which we'll learn more about in Chapter 30...

When antimatter (which we'll learn more about in Chapter 30 ) interacts with an equal mass of ordinary matter, both matter and antimatter are converted completely into energy in the form of photons. In an antimatter-fueled spaceship, a staple of science fiction, the newly created photons are shot from the back of the ship, propelling it forward. Suppose such a ship has a mass of $2.0 \times 10^{6} \mathrm{kg},$ and carries a mass of fuel equal to $1 \%$ of its mass, or $1.0 \times 10^{4} \mathrm{kg}$ of $\mathrm{matter}$ and an equal mass of antimatter. a. What is the final speed of the ship, assuming it starts from rest, if all energy released in the matter-antimatter annihilation is transformed into the kinetic energy of the ship? b. Not only do photons have energy, as you learned in Chapter $25,$ they also have momentum. Explain why, when energy and momentum conservation are both considered, the final speed of the ship will be less than you calculated in part a.

When antimatter (which we'll learn more about in Chapter 30 ) interacts with an equal mass of ordinary matter, both matter and antimatter are converted completely into energy in the form of photons. In an antimatter-fueled spaceship, a staple of science fiction, the newly created photons are shot from the back of the ship, propelling it forward. Suppose such a ship has a mass of $2.0 \times 10^{6} \mathrm{kg},$ and carries a mass of fuel equal to $1 \%$ of its mass, or $1.0 \times 10^{4} \mathrm{kg}$ of $\mathrm{matter}$ and an equal mass of antimatter.
a. What is the final speed of the ship, assuming it starts from rest, if all energy released in the matter-antimatter annihilation is transformed into the kinetic energy of the ship?
b. Not only do photons have energy, as you learned in Chapter $25,$ they also have momentum. Explain why, when energy and momentum conservation are both considered, the final speed of the ship will be less than you calculated in part a.

Key Concepts

Matter-Antimatter Annihilation

This concept involves the process where matter and antimatter meet and annihilate each other, resulting in a complete conversion of their mass into energy. The energy is typically released in the form of high-energy photons. This understanding is central to reactions that appear in high-energy physics and is foundational for applications involving energy conversion at extreme densities.

Conservation of Energy

The principle of conservation of energy states that the total energy in an isolated system remains constant, though it may change forms. In the context of matter-antimatter annihilation, the rest mass energy is converted into electromagnetic energy, which can then be harnessed, in theory, to provide kinetic energy to another object, such as a spacecraft.

Kinetic Energy and Rocket Propulsion

Kinetic energy in mechanics is the energy an object possesses due to its motion. When energy released from annihilation is allocated entirely to convert into kinetic energy of a spaceship, one can use classical or relativistic formulas to determine the resulting speed. This concept is closely related to how energy conversion principles underpin rocketry and other propulsion methods, even if idealized.

Photon Momentum and Conservation of Momentum

Photons, despite having no rest mass, carry momentum, and their emission imparts thrust to a spacecraft through conservation of momentum. When considering both energy and momentum conservation, some energy is necessarily 'used' to account for the momentum of the photons, which means that less energy is available to be converted into the kinetic energy of the ship, leading to a final speed lower than that predicted by energy considerations alone.

Transcript

00:01 Hello everyone.

00:02 In this problem, we're asked to calculate the final speed of a spaceship, given that it carries a certain fraction of its mass as fuel in the form of matter and antimatter.

00:14 And by annihilating these two compounds, they produce energy.

00:19 So first, we're going to assume that all of the energy that comes from the annihilation of matter -antamatter is going to be equal to the kinetic energy of the spaceship.

00:31 And so the way we do that is by assuming, is by, you know, collected the details.

00:37 So the mass of the spaceship is 99 % of 2 times 10 .6, which works out to be 1 .98 times to 6 kilograms, while the mass of the fuel, you know, is equal parts matter and antimatter.

00:50 So it's in total 2 .0 times 10 to 4 kilograms.

00:54 Now, this mass of the fuel is going to be the rest mass of that, or the rest mass energy of that fuel is going to be the kinetic energy of the spaceship.

01:09 So we're going to say that the kinetic energy is mf times c squared.

01:13 And so using the relativistic energy equation for the spaceship, we're going to see that the total energy of the ship is equal to the mass of the ship times its gamma factor, its final gamma factor, times c squared, which if we write this out, this is equal to the rest mass energy, which is just ms times c squared, plus the kinetic energy, which consists of a bunch of other terms, which a first approximation is a half mv squared.

01:45 So writing that out, we know what the actual value of this ek is going to be because we're told that it's going to be mf times c squared.

01:53 So putting that in, we have that.

01:55 The total energy of the spaceship is ms c squared plus m f c squared which is equal to the total mass of ship plus fuel times c squared so that's going to be the total energy and this total energy is going to be equal to the mass of the spaceship times gamma times c squared so the c squares cancel on both sides and we're going to have that if we divide by ms plus mf from the right hand side and divide by gamma from the left hand side, we arrive at ms over ms plus mf is equal to 1 over gamma, which is equal to the square root of 1 minus v squared over c squared, where v is the speed of the spaceship.

02:35 So then what do we do? well, we can just rearrange this for v, and we do that by first square in both sides.

02:42 So ms over ms plus mf squared is equal to 1 minus v squared over c squared, and then we add v squared over c squared to both side and subtract ms over ms plus mf is all squared from both sides...

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