Some solved physics problems
Compute the minimum speed to throw an object between 2 given
points of the surface of a spherical, non-rotating planet.
Take the example of a 90° angle of distance between these points
(the same method can work with any other distance).
Say this is from the pole to the equator.
(I wrote this answer to the above question in a forum, so I copy it
here.)
The minimum speed of launch, means :
The minimum orbital energy
The minimum length of the semi-major axis
But the length of the major axis equals the sum of distances of any
chosen orbital point to both foci.
One focus is the center of the planet.
The pole is an orbital point, its distance to the center of the
planet is equal to the radius R of the planet.
The direction of the major axis is fixed : it makes a 45° angle
with the polar axis.
On this axis we need to choose the other focus at the minimum
distance to the pole. We find it as the orthogonal projection of the
pole on the major axis.
This minimum distance is R/√2.
Conclusion: the minimum length of the semi-major axis is L=
R(1+1/√2)/2.
Orbital energy is proportional to -1/L
The escape velocity V provides the difference of orbital energy
between falling to the center (energy -2/R for a major axis of
length R made of the planet's radius) and escaping (energy = 0 for
an infinite semi-major axis)
Our minimum speed v provides the difference between falling (energy
-2/R) and this trajectory (energy -1/L = -2/(1+1/√2)R).
Thus, an energy of (1 - 1/(1+1/√2) ) = (√2 -1) times the escape
energy.
Take the square root to get the ratio of speeds.
(More problems will be added later. If you have a nice problem, I
might look for a solution and add it here too)
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