# When B is directly below A, as shown, what's B’s velocity (in vector notation) relative to A?

Spacecraft A and B are in coplanar circular orbits around the Earth. The orbital radii are as shown in the figure.

(Not looking to use C-W equation. Just the equation describing relative velocity between 2 bodies.)

• Are you looking for a speed (scalar) or velocity (vector) answer? You use both terms. Nov 21 '20 at 21:28
• The vector, velocity. Sorry about that
– Nash
Nov 21 '20 at 22:13
• Suggest you edit the question and clarify. Nov 21 '20 at 22:16
• @Nash I believe you may be over thinking this question. As others have noted in other comments, simply subtract the velocities. A is travelling at 7053.4 m/s perpendicular to the radius, and B is travelling 7540.4 m/s also perpendicular to the radius. Their directions are the same (if they're right over each other). Thus B's relative velocity wrt A is 487 m/s, west. Nov 21 '20 at 23:11
• If you know how to express the speed of A in vector notation then you know also the speed of B. You only have to know how to subtract two vectors and you are able to answer the question. Can you tell us which step you know and which not?
– Uwe
Nov 23 '20 at 19:10

In a circular orbit, the velocity of a spacecraft is constant throughout, and is computed as follows:

$$v = \sqrt\frac{\mu}{r}$$

Where $$v$$ is the velocity in km/s, $$\mu$$ is the gravitational parameter of the main body (e.g. $$398600.4418~km^2/s^{-2}$$ for the Earth) and $$r$$ is the radius (not altitude) of the spacecraft compared to the center of mass, in kilometers as well.

To compute the relative speed of these two spacecraft, simply compute both of their speeds and subtract them.

• Thanks. But I am looking for a solution in the vector form
– Nash
Nov 21 '20 at 22:14
• @Nash, that's confusing. How are your states defined? In theory, you have a full state vector, either in Cartesian or in Keplerian coordinates. If so, you can compute the state velocity vectors and take their difference. Could you clarify what information you're starting with? Nov 21 '20 at 22:51
• Agreed, just subtract the velocity vectors. There's nothing special about the bodies being in space. Nov 21 '20 at 22:53
• I've seen a similar question but for relative acceleration. The answer in that case is (a_rel)xyz =−0.268i (m/s2). i is a unit vector. Hence, I was hoping to know how can one determine the relative velocity (v_rel) in a similar state vector form. I believe it'd be using: vB = vA + omega × r_rel + v_rel but I'm facing difficulty with the same since it's my first time
– Nash
Nov 22 '20 at 0:08