Question

When antimatter 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 2.00×10^6kg, and carries a mass of fuel equal to 4% of its mass, or 4.00×10^4kg of matter and an equal mass of antimatter.

Required:
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?

Answer 1

v = 5.88 10⁷ m / s

Step-by-step explanation:

For this exercise we use the relation

E = m c²

also indicate that all energy is converted into kinetic energy

E = K = ½ (M-2m) v²

where m is the mass of antimatter and M is the mass of the ship's mass. Factor two is due to the fact that equal amounts of matter and antimatter must be combined

we substitute

m c² = ½ (M-2m) v²

v² =
`2 (m)/(M+2m) \ c^2`

let's calculate

v =
`\sqrt{2 \ (4 \ 10^4 )/(2 \ 10^6 + 2 \ 4 \ 10^4) \ (3 \ 10^8)^2}`

v =
`\sqrt{ 34.615 \ 10^(14)}`

v = 5.88 10⁷ m / s