Howcome Accelerating In Space Make You Go Slower?

Question

We all know that objects at a lower orbit have a higher speed and objects at a higher orbit have lower speed. If you're a spacecraft trying to get from a lower orbit to a higher orbit, you would accelerate two times.
Once at your periapsis, burning prograde.
Then once at apoapsis, burning prograde, to raise your periapsis.

Any school student can tell you doing two accelerations in your direction of travel will obtain you a higher linear speed. So how is this possible?

I did some math. Say you're at Kerbin with an 85km circular orbit. You want to get to a 100km circular orbit. Orbital speed at 85km is 2270 m/s. Orbital speed at 100km is 2246 m/s. By the vis-viva equation, speed at periapsis is 2283 m/s, speed at apoapsis is 2234 m/s. 1st burn: 2283 - 2270 = +13 m/s. 2nd burn: 2246 - 2234 = +12 m/s. How do you explain what's going on?

Answer 1

Any school student can tell you doing two accelerations in your direction of travel will obtain you a higher linear speed.

The word "linear" is the cause of your problem.

Orbital motions are not linear.

Say you're at Kerbin with an 85km circular orbit.

We've a perfectly good real planet called Earth, why use a fake one ? :-)

You want to get to a 100km circular orbit.

The classic transfer orbit is a Hohmann transfer. The detailed maths is explained on that page. Let's just get to the concept here.

When you're in a circular orbit, your orbital velocity must be constant and relates to your orbital radius.

If you apply a delta-v to increase your velocity at any time, you change the orbital path you must follow from circular to elliptical.

Now the thing about an elliptical orbit is that your velocity is not (repeat not ) constant.

At closest approach you are at highest velocity and at furthest, you are at the slowest velocity. You can think of this as gaining speed as you fall closer and loosing it as you climb away.

So in this transfer you add speed to the object in the circular orbit and it enters an elliptical orbit. When it reaches the "top" of it's orbit (which we aim for by choosing the delta-v carefully), we are traveling more slowly.

The velocity we need for a circular orbit at this point is higher than the velocity we have in the elliptical orbit we are currently in at that point, so we must (again !) add delta-v to reach that velocity.