Why does not the Moon get faster?

Question

I've learned that an object has an equivalent acceleration motion due to gravity in a vertical direction in parabolic motion. And the moon is also an example of parabolic motion, right? As Isaac Newton said, I mean, the cannon one. I wonder why the Moon doesn't get faster when it is falling to the Earth?

Answer 1

... the moon is also an example of parabolic motion, right?

Not quite. The Moon's orbit relative to the Earth is (approximately) an ellipse. Like a parabola, an ellipse is a conic section. Unlike a parabola, it is a closed curve.

I wonder why the moon doesn't get faster when it is falling to the Earth?

The Moon does travel faster when its orbit brings it closer to the Earth, and it travels slower when it is further away from the Earth. However, its orbit is quite close to being a circle - the difference between its closest approach to Earth (perigee) and its furthest distance from Earth (apogee) is only about $12\%$ of the average radius of its orbit. So you would have to take some quite precise measurements to detect this effect.

Answer 2

I have not studied about orbit formulas of orbital velocity yet, but you can rely on the following before going to orbit related formulae. A simple way to understand is that if the moon was closer enough to the earth, would it fall? Yes. Why, in fact even a ball if set with a perfect velocity (in the way the moon was set after formation) far from earth, will of course act like artificial satellites (which act like moon). Your problem here is: 1. Moon does not have a parabolic path. 2. Acceleration due to gravity is unable to bring down moon, which is why the above statement holds true.

The acceleration due to gravity acting on the moon at any time instant is acting perpendicular to its direction of motion, hence unable to change its tangential velocity vector. It is this accurate velocity of nature's creation that allows moon to balance its orbit without falling to earth (the fact that it is going farther away with years, is also true). Why so? Because the acceleration due to gravity has 0 as its horizontal component, hence unable to put a non zero tangential (horizontal) acceleration on the moon.

But the moon's orbit isn't perfect circle, slightly different. Hence earth is not at centre of this orbit, thus leading to a point along the orbit (likewise the orbit of earth around sun), where the moon is a bit closer to earth than the opposite point.

The graph of y(x) coordinate of moon, by tracing its orbit on a plane, gives an ellipse with very little separation of focus and centre (nature's creation). On any elliptical orbit there are those 2 points. And obviously by common sense, acceleration due to gravity which depends only on r² ($g=GM/r²$ where r is distance between centres of gravity or centres of spherical mass of the 2 masses, G, M are constant), will change logarithmically as the square of the distance of separation. Since there exists such 2 points, near the point where the moon is closer, it must be fastest, and slowest near the point where it is farthest. All because instead of a circular orbit, the moon has an elliptical orbit. But the difference in speed is not much.

Another way to address the 1st question is through mathematics: once we can show that moon goes along elliptical orbit, we can show that it's path cannot be elliptical because: we know, no part of an ellipse (generally) is a parabola. There is also a mention by another person here, that the orbit is a closed curve like ellipse, whereas parabola is not closed.

Hope I cleared some doubts today.