• Geofleur

#### Geofleur

Gold Member
I have been reading D.E. Littlewood's book "The Skeleton Key of Mathematics", and near the beginning he says that if a projectile weighing one (long) ton were given a velocity of 44 miles a second, this would be "sufficient to raise it to a height of 1000 miles above the Earth's surface."

Naturally, I wanted to reproduce this number for myself, so I started with conservation of energy:

## \frac{1}{2}M_P V^2 - \frac{GM_P M_E}{R_E} = -\frac{GM_P M_E}{R_E+h} ##,

where ## M_E ## is the mass of the Earth, ## M_P ## that of the projectile, ## h ## is the height of the projectile above the surface, ## G ## is the gravitation constant, and ## V ## is the projectile's speed. Solving for the height gives

## h = \frac{R_E}{\frac{2GM_E}{V^2 R_E}-1} ##.

First, note that the mass of the projectile does not appear in this formula. It appears, then, irrelevant that the projectile weighs a ton. Second, as the denominator approaches zero, ## h \rightarrow \infty ##; setting the denominator to zero and solving for ## V = V_{esc} ## yields

## V_{esc} = \sqrt{2G M_E / R} \approx 7 \frac{mi}{s} ##.

Thus, 44 mi/s seems like way more than you would need to lift the projectile 1000 miles above the surface. I didn't account for energy lost to air resistance in these calculations, but launching an object through the atmosphere into space like that seems unfeasible - wouldn't it just burn up? Am I missing something or is Littlewood's calculation off?

Well done - if you google "escape velocity in miles per second" what do you get?

Well, Google didn't give me what I expected, but Wolfram Alpha did - about 7 mi/s. For some reason, I always have a hard time believing that books could be wrong about things like this. I always think I must be missing something. Thanks!

<snip> I didn't account for energy lost to air resistance in these calculations, but launching an object through the atmosphere into space like that seems unfeasible - wouldn't it just burn up?

Uh... yeah. A 1-ton projectile launched at Mach 208 is going to create significant atmospheric disturbances. Spacecraft coast to Earth at a relatively plodding Mach 25, and how'd that work out for Challenger? Even slender little bullets lose most of their kinetic energy after a few hundred yards:

https://en.wikipedia.org/wiki/Ballistic_coefficient#/media/File:Effect_of_BC_on_Energy_Retained.jpg

• Geofleur