Question about orbits of satellites.

Satellites are constantly accelerating, but the acceleration is towards the center of the Earth causing the satellites to turn. The instantaneous linear velocity of the satellite remains constant though.

You left out the way gravity works. At a certain distance from the earth the mass of an object and its orbit are “held” at an always-falling, non accelerating rate as the earth turns from under it. Attitudinal changes can turn a stable orbit into a spiral, and eventually the object will burn up from friction in the atmosphere, or impact the earth.

Think of it as the tradeoff between kinetic energy ½ mV^2, and potential energy (against gravity),. mgh. If you try to go faster you will move to a higher orbit. You cannot keep gaining speed without getting the energy from somewhere.
Actually there are all kinds of tricks that can be played when you are moving from one gravity field to another i.e. sling shotting to accelerate. The interplanetary probes do this. But your example has you going around the earth so you can ignore that for now.

The speed with which they move around the earth is not their falling speed. It is the speed with which they are trying to get away from earth. The gravitational force exerted upon them by the planet acts counter to their initial movement vector to veer them of their path, into their eventual orbit. An orbiting objects speed, mass and distance are aligned in a way that gravity only manages to hold them at an almost steady distance, so that their actual falling speed towards earth is actually almost zero.
Of course real satellites are actually falling, just very slowly. As is the ISS. Their orbit is not high enough for their speed to cancel out gravity completely, so gravity has a slight edge and pulls it slowly towards earth.
The moon is the opposite case. Its orbit is too high, resulting in it moving slowly away from earth, a negative falling speed, so to speak.

@ragingloli
Thanks I get it now I thought it was that I just wanted it fixed within my head.

@quilm I read the other answers. I want to add my own. Consider a satellites that flies in a “circular” orbit that passes above the North Pole and South Poles. When the satellite is passing above the North Pole it is “falling” towards the South Pole just as if you dropped an apple in the snow on the North Pole it would be a few feet closer to the South Pole. Its southward speed is increasing. However, when the satellite is above the South Pole it is falling towards the North Pole. Its northward speed is increasing! Notice that in a full orbit the northward speed and the southward speed both increase by the same amount cancelling each other out. In this fashion the satellite is always accelerating but never increases its velocity relative to earth’s velocity.

@ragingloli: “As is the ISS. Their orbit is not high enough for their speed to cancel out gravity completely, so gravity has a slight edge and pulls it slowly towards earth.” <== this isn’t true! If you put the ISS inside a totally evacuated tube it would stop falling. Friction is bringing it down. You could even build such a tube at ground level and we could orbit things indefinitly at sea-level. (*** ok, there is more than air-friction such as induction, etc but these are not the forces that brought down skylab or that will bring down the ISS)

Just one more question do satellites go in ellipses or circles around the earth?

And if ellipses doesn’t that mean they are falling?

@dotlin Yes, they go in an ellipses.

No, moving in an ellipse does not mean something is “falling”. A race-car could go in an ellipse if that was the way the track was shaped. Maybe the object is traveling in an ellipse because of a string, etc. There are lots of reasons possible other than orbiting.