In the 1950s, a series of projects, collectively called Project Orion were propo

Question

In the 1950s, a series of projects, collectively called Project Orion were proposed to utilize Nuclear Explosions to move a rocket into space. This research was later ended in 1963 by the Partial Nuclear Test Ban Treaty, which was signed by the US, USSR and UK governments. Because of the tremendous amount of energy released by a fission (or fusion) reaction, this project had the potential to reduce the cost of space travel, albeit at the expense of massive radioactive fallout on Earth.

What fraction of the rocket's mass would need to be converted into energy to get it into Low Earth Orbit (LEO)? Neglect the decrease in gravity between ground and orbit but be sure to include the calculation for your increase in speed!

If you include the decrease in gravity, how much less mass is required as a fraction of the total rocket?

If your rocket is initially 1.00

Explanation / Answer

initial v = 2 pi R/T

now need to find final speed

G M m/r^2 = mv^2/r

v^2 = G M/r

v = sqrt( G M/r)

now conservation of energy
Ei + dE = Ef
1/2 mvi^2 - G M m/r + xm c^2= 1/2 ( m - x m) v^2 - G M (m-xm)/r

1/2 vi^2 - G M/r + x c^2 = 1/2 ( 1- x) v^2 - G M (1-x)/r

1/2 ( 2 pi R/T) - G M/r + x c^2 = 1/2 (1-x) G M/r - G M (1-x)/r = - 1/2 (1-x) G M/r

0.5*2*pi*6.37E6/(24*60*60)^2 - 6.67E-11*5.97E24/6.37E6 + x*3.0E6^2 = -0.5*(1-x)*6.67E-11*5.97E24/6.37E6

x=3.47E-6

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