Find delta-v; Hohmann transfer orbit

[oddjobmj]

Homework Statement

A space vehicle is in circular orbit about the earth. The mass of the vehicle is 3300 kg. The radius of the orbit is 2RE. It is desired to transfer the vehicle to a circular orbit of radius 4RE.

An efficient way to accomplish the transfer is to use a semielliptical orbit (known as a Hohmann transfer orbit), as shown. What velocity changes are required at the points of intersection, A and B? Express the change in speed as the percentage of the speed just before the change.

Unfortunately I am not given the figure that this problem refers to.

Homework Equations

Δv₁=\sqrt{GM/r_1}(\sqrt{2r_2/(r_1+r_2)}-1)

Δv₂=\sqrt{GM/r_2}(1-\sqrt{2r_1/(r_1+r_2)})

v=\sqrt{GM(2/r-1/a)}, where a is the semi-major axis.

The Attempt at a Solution

I thought this was pretty straightforward but apparently my answers are incorrect. I'm not sure what I'm doing wrong. My interpretation is that the satellite is in orbit at 2R_E=r₁, we need to get it into a transition orbit and to do so we increase the velocity by amount Δv₁, we then stabilize this orbit at 4R_E=r₂ by increasing velocity once more.

Plugging in the knowns to the above equation I get:

Δv₁=865.2 m/s

Δv₂=725.7 m/s

Now I need to represent these changes in velocity as percentages of the velocity just before the given change. The initial velocity can be found using this equation:

v₀=\sqrt{GM(2/r-1/a)}

In this case, being a circular orbit, r=a=2R_E

v₀= 5592.8 m/s

Also, the velocity after the first velocity change is v₁=v₀+Δv₁=6458 m/s

So, the first answer should be:

\frac{Δv_1}{v_0}=15.47%

and the second answer:

\frac{Δv_2}{v_1}=11.24%

I also tried entering these as the decimal equivalents and tried simply using the initial velocity as the divisor when calculating my percents. None of this worked. Perhaps I have A and B mixed around because I don't have the image but I believe it is more likely that I made a mistake.

These answers are wrong. Any suggestions? Thank you!

EDIT: The reason I am given the extra variables in the initial problem is that there was a part before this that asked me to calculate the required energy to make the transition from 2R to 4R which I solved correctly.

 

[gneill]

The velocity just before the second change will not be the same as the velocity immediately after the first change; the craft is at perigee immediately after the first change, and heads out to apogee where the second change occurs. Otherwise your calculations appear to be fine.

 

[oddjobmj]

Ahh, of course! Thank you, gneill!

Would I be able to plug in 4R_E to the velocity equation? I ask because the satellite's trajectory is skewed in the sense that when it reaches 4R_E it would not be stable if not for Δv2. I'm not sure how to figure the velocity at that point otherwise.

 

[gneill]

Ahh, of course! Thank you, gneill!

Would I be able to plug in 4R_E to the velocity equation? I ask because the satellite's trajectory is skewed in the sense that when it reaches 4R_E it would not be stable if not for Δv2. I'm not sure how to figure the velocity at that point otherwise.

Your velocity equation holds for all points on the transfer trajectory, which would just be an ellipse if not for the second correction. So sure, plug in 4RE and obtain the velocity.

 

[oddjobmj]

Ah, I see what you mean, thank you. My thought initially was to use 4RE assuming the orbit was circular. Of course, the transfer orbit is, as you mentioned, an ellipse. For anyone who has a similar problem in the future you can figure the semi-major axis of this ellipse to use in the velocity equation from the transfer orbit's perigee and apogee points which happen to fall on the two known circular orbits.

Using your suggestion(s) and that noted above the correct answer was found.