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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?
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?