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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.)

enter image description here

Nash
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  • Are you looking for a speed (scalar) or velocity (vector) answer? You use both terms. – Organic Marble 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. – Organic Marble 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. – Star Man Nov 21 '20 at 23:11
  • 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 00:07
  • I don't know what specific equation or formula you're referring to. I don't know what omega is supposed to represent. However calculating relative velocities where both objects are travelling in the same direction is simply a matter of subtracting them. If you want to represent the answer in a specific form, then you should include that in your question. – Star Man Nov 22 '20 at 00:35
  • And judging by the equation that you wrote, I assume omega means angle between the two objects? If so, then the middle term in $V_B = V_A + \omega \times R_{rel}+V_{rel}$ will equal 0. Therefore it could be written as $V_{rel} = V_B - V_A$ – Star Man Nov 22 '20 at 00:44
  • that's the equation of relative velocity b/w 2 bodies. And omega is the angular velocity Thanks for the help though. I'll take it up with my prof. after the weekend is over. – Nash Nov 22 '20 at 16:12
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    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

1 Answers1

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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.

ChrisR
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  • Thanks. But I am looking for a solution in the vector form – Nash Nov 21 '20 at 22:14
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    @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? – ChrisR Nov 21 '20 at 22:51
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    Agreed, just subtract the velocity vectors. There's nothing special about the bodies being in space. – Organic Marble 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 00:08