Final answer:
Earth's velocity of approach toward the Sun at an extremum of the latus rectum can be approximated using the average orbital velocity of the Earth, which is approximately 30 km/s.
Step-by-step explanation:
To calculate Earth's velocity of approach toward the Sun when Earth is at an extremum of the latus rectum through the Sun, we need to consider Earth's orbit around the Sun and apply Kepler's laws of planetary motion. The latus rectum of an ellipse refers to the line that is perpendicular to the major axis and goes through one of the foci. At these points, also known as the vertex of the ellipse, Earth's velocity is perpendicular to the line joining it to the Sun, which means all of Earth's velocity at this instant is directed towards or away from the Sun, i.e., in the radial direction.
According to Kepler's second law, also known as the law of equal areas, the line joining the Sun and Earth sweeps equal areas during equal intervals of time. However, the speed is not constant due to the elliptical nature of the orbit. The speed is highest when the Earth is closest to the Sun (perihelion) and lowest when it is farthest (aphelion). To find the actual speed of the Earth at the latus rectum, we can use the fact that the orbital velocity of the Earth around the Sun is approximately 30 km/s. This value can be taken as a good approximation for the speed at the extremum of the latus rectum, as it averages the velocities at aphelion and perihelion.