Fireing projectile from north pole to equator

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SUMMARY

The discussion centers on determining the necessary launch angle and velocity for a projectile fired from the North Pole to land on the equator. A proposed angle of 45 degrees with a muzzle velocity of 9401 m/s was deemed impractical. The conversation highlights the complexities of the problem, emphasizing the need to account for Earth's curvature and gravitational force, which decreases with distance from the center of the Earth. The consensus is that achieving this trajectory requires a horizontal launch at a velocity close to that of a circular orbit, while neglecting air resistance and Earth's surface irregularities simplifies the calculations.

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  • Knowledge of orbital mechanics and circular motion
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greaser
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I am trying to find what velocity and angle of launch is required for a projectile to be fired from the North pole and land somewhere on the equator. I was thinking 45 degrees with muzzle velocity 9401m/s but that sounds ridiculous. Also how much time would it be in the air for?
 
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Looks to me like a very difficult problem, probably involving solving a nasty non-linear differential equation.

IF it were a matter of finding the angle and initial velocity of a flat plane, with constant gravitational acceleration, a distance equal to the distance from the north pole to the equator, then it would be relatively simple. However, if you REALLY mean "north pole to the equator" you will have take into account the curvature of the Earth and the fact that you are firing the projectile around 1/4 of that curvature. Your trajectory will be high enough that you will have to take into account the fact that the gravitational force falls off as 1/r2.
 
I think this problem becomes simple if we may neglect air friction and any deviations of the Earth's surface from being a sphere.
Because in this case, an orbit will do which is 'infinitesimally close' to a circular orbit. Which means: fire horizontally with circular orbit velocity.

I'm afraid we must *not* neglect those...
 

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