The plane example probably isn't the best because any plane you're going to see is small from ones perspective.
Try to visualize a large sphere passing over you. Now imagine what happens to that sphere as it continue moving over a flat plane. It will get smaller and smaller until it disappears from view. That disappearing isn't instant, it's a gradual shrinking until it's out of sight.
Example
You staring at the sun at 1pm (pic8.co)
You staring at the sun at 3pm (pic8.co)
Notice how the angle of ascent from the ground to the sun decreases at the sun moves away? The sun stayed moving straight along a flat plane, and yet from your perspective the sun moved DOWN towards the horizon.
Hope that helps
Think I'm going to make that drive again.
You may find this interesting (vid8) if you're familiar with Pixar, DIsney, and are intrigued by arguments against globe earth theory.
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