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- How long does it take for the Earth to fall half the distance to the Sun, without calculus?

There's a classic physics problem that is:

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take to fall into the Sun (neglecting the size of the Sun/Earth)?

I know that a clever way to solve this problem is by using degenerate ellipses and an object "falling" into the Sun is the equivalent of the object orbiting the Sun with very high eccentricity. Halving the semi-major axis of the orbit and using Kepler's Law gives: T=1AU/(4√2). However, I want to apply to strategy to another similar problem. This time:

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take for the Earth to travel half the distance to the Sun(neglecting the size of the Sun/Earth)?

If we were to use the result for T from the first problem and then simply apply Kepler's 2nd Law by looking at ratios of areas, we should be able to find this new time it takes to fall halfway to the Sun. However, I am having a problem visualizing what area of the ellipse the Earth covers when it travels half the distance to the Sun. I know about the calculus solution which is really bashy, so I would like some assistance with figuring out this more elegant method.

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take to fall into the Sun (neglecting the size of the Sun/Earth)?

I know that a clever way to solve this problem is by using degenerate ellipses and an object "falling" into the Sun is the equivalent of the object orbiting the Sun with very high eccentricity. Halving the semi-major axis of the orbit and using Kepler's Law gives: T=1AU/(4√2). However, I want to apply to strategy to another similar problem. This time:

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take for the Earth to travel half the distance to the Sun(neglecting the size of the Sun/Earth)?

If we were to use the result for T from the first problem and then simply apply Kepler's 2nd Law by looking at ratios of areas, we should be able to find this new time it takes to fall halfway to the Sun. However, I am having a problem visualizing what area of the ellipse the Earth covers when it travels half the distance to the Sun. I know about the calculus solution which is really bashy, so I would like some assistance with figuring out this more elegant method.