# 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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