All Problems Universal Gravitation
Problem 1.203
A small body starts falling onto the Sun from a distance equal to the radius of the Earth's orbit. The initial velocity of the body is equal to zero in the heliocentric reference frame. Making use of Kepler's laws, find how long the body will be falling.
1.203. Falling of the body on the Sun can be considered as the motion along a very elongated (in the limit, degenerated) ellipse whose major semi-axis is practically equal to the radius \(R\) of the Earth's orbit. Then from Kepler's laws, \((2 \tau / T)^{2}=[(R / 2) / R]^{3}\), where \(\tau\) is the falling time (the time needed to complete half a revolution along the elongated ellipse), \(T\) is the period of the Earth's revolution around the Sun. Hence, \(\tau=T / 4 \sqrt{2}=65\) days.