IOAA 2021, Теория — Задача 11. Minimum velocity of a projectile (15 points).

• 11.1 7195 m/s
• 11.2 $e=1/(1+\sqrt2)=0.414$

11.1

[1 point]

[1 point]

At points A and B, the $v$ is the same; so, if $v$ is minimum, EM is minimum:

We know that the mechanical energy for an elliptical orbit is given by:

[1 point]

where 2a is the length of the major axis.

Since EM must be minimal, then 2a must be minimal [1 point]

[1 point]

O: Center of the Earth and one of the focus of the ellipse.

F: the other focus of the ellipse

[1 point]

$OB+BF=2a$, must be minimal.

F takes an arbitrary position, so to minimize, BF must be perpendicular to OF:

[2 points]

[1 point]

[1 point]

This is the minimum value it can take

Correspondingly, the expression for the minimum velocity is

[1 point]

Using the M and R values for the Earth, the minimum velocity is

[1 point]

11.2

Furthermore, $OF=2e\cdot a$, where $e$ is the eccentricity

[2 points]

[1 point]