A planet or asteroid the sun in an elliptical path. The diagram shows two points of interest: point P, the perihelion, where the planet is nearest to the Sun, and the far point, or aphelion, at point A. At these two points (and only these two points), the velocity v of the planet is perpendicular to the corresponding "radius" vector , from the Sun to the planet. An arbitrary point Q is also shown.
Suppose the asteroid's speed at aphelion is , and its distance from the sun there is (AU is the "astronomical unit", commonly used in planetary astronomy).
a) Determine the planet's speed at perihelion.
b) Determine the planet's distance from the Sun (in AU) at perihelion.
Assume that the Sun is orders of magnitude more massive than the asteroid. This means the Sun does not accelerate significantly in response to the planet's gravity and can be assumed to remain at rest.
Derive a symbolic formula, but don't try too hard to simplify it. Look up the values and conversion factors you need in order to obtain a numerical answer.
Recall the universal gravity formulas:
Force
potential energy
Consider the Sun-asteroid system to be isolated and apply conservation of energy.
