The Jules Verne Problem
A projectile is placed in a Columbiad pointed at the zenith. This projectile is to be launched to the moon at a time where it will arrive at the moon at the moment the moon is full.
What is the minimum initial velocity, , in yards per second, that can be given to this projectile so that it can avoid falling back to the Earth? Give answer to the nearest thousand.
Givens
Universal gravitational constant ( ) = -
Earth's mass ( ) =
Moon's mass ( ) =
Projectile's mass ( ) =
Earth's radius ( ) =
Distance from Earth's center to moon's center ( ) =
Assumption: The projectile loses exactly 1/3 of its initial velocity traversing the Earth's atmosphere (i.e., if its initial velocity was 3,000 ft/sec, its velocity upon leaving Earth's atmosphere = 2,000 ft/sec)
Assumption: The projectile survives the initial blast from the Columbiad intact.
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1 solution
Let H ( t ) be the height of the projectile above the Earth's surface at time t . where t is the amount of time since the detonation of the Columbiad.
H ( 0 ) = 0 , H ′ ( 0 ) = V
Force pulling towards the moon = G m p / ( D − R − H ( t ) ) 2
Force pulling towards the Earth = G M p / ( R + H ( t ) ) 2
So H ′ ′ ( t ) = G m p / ( D − R − H ( t ) ) 2 − G M p / ( R + H ( t ) ) 2
The point of no return, where the projectile must fall to the moon instead of the earth, occurs when H ′ ′ ( t ) = 0 .
Solving, we find that this point is at a distance N = D / ( 1 + m / M ) = 1 . 1 1 3 ∗ 1 0 9 f e e t
To reach this point, the velocity V , excluding air resistance, has to be 12,105 yards per second. So multiply this by 3/2 to account for air resistance, and you get V = 1 8 , 1 5 4 yards per second.