Originally Posted by

**galexander**
But why does this ‘centripetal acceleration’ not add during each interval of time to the orbital velocity using vector addition and Pythagoras’ Theorem?

It does.

Think of the satellite doing one half of the orbital circle. For convenience, and so I don't have to make drawings, let's imagine a huge clock face filling this circle, and suppose the satellite goes from the twelve o'clock mark to six (in the, erm, clockwise direction).

Now take all those tiny arrows, say a million of them, pointing inwards from the orbit. The one at 12 points downwards, the one at 1 points down and slightly left and so on, the one at 3 points left, the one at 4 points left and somewhat up.... If you move them around (by pure translation, no turning!) so they are arranged nose to tail, they will form a half-circle shaped like the bottom half of our clock face.

So the sum total of those tiny arrows is one diameter of the circle, pointing left.

Remember that these are velocity vectors, not positions of the satellite. Their dimensionality is in m/s, and the radius of the half-circle they form is equal to the orbital speed. The diameter is, of cource, twice that much.

Now while the

*speed *of the satellite is constant (as a scalar), its

*velocity *(vector) changes dramatically. At the 12 o'clock mark, the satellite was moving right. At 6, it's going left. The change in velocity over this time is a vector 2v long, pointing left - precisely what we found as the sum of the little arrows.

After a full circle, the total change in velocity is zero.

Does this help?

(note: summing a million tiny arrows is an approximation. The proper way to do this is integration - that is, summing infinitely many infinitely small arrows)