From the equator to the North Pole
- Round
- Theory
- Points
- 15
- Level
- Silver
- Taught in
- Section 3.3
Adapted from IOAA 2021 (Bogotá, held online), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.
(a) What is the smallest speed at which a projectile must be launched from Earth's surface at the equator so that it reaches the North Pole? [12 points]
(b) What is the eccentricity of the projectile's path? [3 points]
Ignore Earth's rotation and its air, and take Earth to be a sphere.
- The path is an ellipse with Earth's centre O at one focus, through the launch point L and the North Pole N, which are both a distance
- F is therefore equally far from L and N, and the two distances together are at least the length of the chord LN, which is
- The smallest
- By (3.5),
- The distance between the foci is
- An ellipse makes equal angles at each point with the lines to its two foci, so the line at right angles to the path at L halves the angle between LO and LF. LO points straight down and LF runs along the chord at 45° to it, so that line is 22.5° from the vertical, and the projectile leaves at 22.5° above the horizontal.
AnswerThe smallest speed is 7.20 km/s, launched at 22.5° above the horizontal towards the north, and the path is an ellipse with eccentricity
This problem also appears among the olympiad problems of Section 3.3, The shape of an orbit, which teaches the ideas it uses. Equation and figure numbers refer to the course.