I was taught that when the acceleration experienced by a body is constant, that body follows a parabolic curve. This seems logical because constant acceleration means velocity that is linear and position that is quadratic. This is what I learned from projectiles: Bodies are thrown with an initial velocity near the surface of the Earth, they experience constant acceleration and the result is a parabolic curve.
Now that doesn't apply to the orbit of the Earth. The gravitational force can be thought of as constant since the distance from the Earth to the Sun can be thought of as constant too, which by Newton's Second Law means the acceleration of Earth is also constant. Wouldn't that mean that the Earth should just follow a parabolic path?
Is there a mathematical proof (similar to the one I mentioned about projectiles) giving the elliptical orbit as a result?
My question is, in a word, why can't the Earth be treated as a projectile? And if it can then why doesn't it behave like one?
Answer
Now that doesn't apply on the orbit of the Earth. The gravitational force can be thought of as constant since the distance fron Earth to Sun can be thought of as constant too
You are correct that the strength or magnitude of the sun's gravitational field is very similar over the length of the earth's orbit, but the direction is not. In a uniform gravitational field, the direction would be the same everywhere.
Over the path of the earth's orbit, the sun's gravitational field points in different directions. This significant difference from a uniform field means that the earth's orbit is quite far from a parabola.
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