Consider an interstellar spaceship powered by matter-antimatter reactions. The energy resulting from the matter-antimatter reactions is ejected from the exhaust as electromagnetic radiation producing thrust. The design goal is to have the thrust produce an acceleration equal to the acceleration of gravity at the earth's surface, g, so the astronauts will experience the effect of normal gravity as they travel. If the spaceship has an initial mass (including fuel) of 40.0 Mg (1000 kg), what is the magnitude of the rate of matter-antimatter burning (in kg/s) to produce the desired acceleration?

Question

Consider an interstellar spaceship powered by matter-antimatter reactions. The energy resulting from the matter-antimatter reactions is ejected from the exhaust as electromagnetic radiation producing thrust. The design goal is to have the thrust produce an acceleration equal to the acceleration of gravity at the earth's surface, g, so the astronauts will experience the effect of normal gravity as they travel. If the spaceship has an initial mass (including fuel) of 40.0 Mg (1000 kg), what is the magnitude of the rate of matter-antimatter burning (in kg/s) to produce the desired acceleration?

Answer 1

To find the magnitude of the rate of matter-antimatter burning (in kg/s) required to produce the desired acceleration, we can use the principle of conservation of momentum.

According to Newton's second law, force (F) is equal to the mass (m) multiplied by the acceleration (a). In this case, we want the force produced by the spaceship's thrust to be equal to the force of gravity (mg), so we have:

F = mg

Since we want the acceleration (a) to be equal to gravity (g), we can rewrite the equation as:

F = ma = mg

Now, we need to find the mass of the spaceship at any given moment during the burning process. Assuming the fuel is consumed at a constant rate, we can express the mass (m) of the spaceship as a function of time (t):

m(t) = m0 - t * r

Where:
m(t) -> mass of the spaceship at time t
m0 -> initial mass of the spaceship, including fuel
t -> time elapsed since the start of burning
r -> rate of matter-antimatter burning (kg/s)

Now, we can substitute this expression in the equation for force:

F = m(t) * a = (m0 - t * r) * g

We want this force to be equal to mg, so we can set up the following equation:

(m0 - t * r) * g = mg

Canceling out g on both sides of the equation gives us:

m0 - t * r = m

Now, we can solve for t in terms of the known variables:

t = (m0 - m) / r

Substituting back into the equation for m(t), we can write:

m(t) = m0 - ((m0 - m) / r) * r

Simplifying this equation gives us:

m(t) = m0 - (m0 - m)

Now, we can substitute this value for m(t) back into the equation for force:

F = (m0 - (m0 - m)) * g = g * m

Since this force is produced by the thrust of the spacecraft, it is equal to the rate of matter-antimatter burning multiplied by the velocity of the ejected matter (v):

F = r * v

Now, we can equate this equation for force with the equation g * m:

g * m = r * v

Simplifying the equation, we get:

r = (g * m) / v

Finally, substituting the given values:
m = 40.0 Mg (1000 kg)
g = 9.8 m/s^2 (acceleration due to gravity)
v = speed of the ejected matter (unknown)

We can calculate the magnitude of the rate of matter-antimatter burning (r) by solving the equation:

r = (g * m) / v

However, without knowing the speed of the ejected matter (v), we cannot determine the exact value of r. Additional information or assumptions about the speed of the ejected matter is necessary to find the magnitude of the rate of matter-antimatter burning.