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Okay, let me clarify this isn't for homework or anything. Purely something I thought of when I was driving home. I am not sure if there is already a solution to this problem, there probably is but I want to see how correct my solution is.

The problem is the following. Suppose I am traveling in a spaceship and I am passing by some planet. I want to know the rate of change of theta (angular velocity) of my orientation to this planet as I pass it. You can say I am "above the planet" in the sense that if I pass and I look some angle below my horizontal I see it. The ultimate goal is to calculate how far the planet will change my directional velocity. If you could also give some input about how this relates to the slingshot effect.

Here is a picture to help illustrate what I am saying.

Here is my approach

This spaceship only has an x direction initial velocity so

Vx = Vo + ATcosѲ = Vo + GMTcosѲ/ R^2

Vy = ATsinѲ = GMTsinѲ/ R^2

Let us suppose a change in velocities. For the purpose for the calculation we will neglect how R changes with time.

ΔVx = Vo + GMΔTcosѲ/ R^2

ΔVy = GMΔTsinѲ/ R^2

Now a consider

tan(ΔѲ) = (ΔVy/ΔVx) = ((GMΔTsinѲ/ R^2)/(Vo + GMΔTcosѲ/ R^2))

This reduces to

tan(ΔѲ) = ((GMΔTsinѲ)/ (VoR^2 + GMΔTcosѲ))

Solving for theta and allow ΔT to be change in one second.

ΔѲ = arctan((GMsinѲ)/ (VoR^2 + GMcosѲ))

since ΔT is equal to one we can say that ΔѲ = w (radpersec)

So the final equation becomes

w = arctan((GMsinѲ)/ (VoR^2 + GMcosѲ))

This makes sense to me in many ways because the only way w goes to zero is if Vo or R go to infinity or if Ѳ is initially zero which would just means the object is going straight towards the planet. Also notice, if Vo is zero the object will just approach the planet at Ѳ which also makes sense.

Let me know what you guys think. Thanks.

The problem is the following. Suppose I am traveling in a spaceship and I am passing by some planet. I want to know the rate of change of theta (angular velocity) of my orientation to this planet as I pass it. You can say I am "above the planet" in the sense that if I pass and I look some angle below my horizontal I see it. The ultimate goal is to calculate how far the planet will change my directional velocity. If you could also give some input about how this relates to the slingshot effect.

Here is a picture to help illustrate what I am saying.

Here is my approach

This spaceship only has an x direction initial velocity so

Vx = Vo + ATcosѲ = Vo + GMTcosѲ/ R^2

Vy = ATsinѲ = GMTsinѲ/ R^2

Let us suppose a change in velocities. For the purpose for the calculation we will neglect how R changes with time.

ΔVx = Vo + GMΔTcosѲ/ R^2

ΔVy = GMΔTsinѲ/ R^2

Now a consider

tan(ΔѲ) = (ΔVy/ΔVx) = ((GMΔTsinѲ/ R^2)/(Vo + GMΔTcosѲ/ R^2))

This reduces to

tan(ΔѲ) = ((GMΔTsinѲ)/ (VoR^2 + GMΔTcosѲ))

Solving for theta and allow ΔT to be change in one second.

ΔѲ = arctan((GMsinѲ)/ (VoR^2 + GMcosѲ))

since ΔT is equal to one we can say that ΔѲ = w (radpersec)

So the final equation becomes

w = arctan((GMsinѲ)/ (VoR^2 + GMcosѲ))

This makes sense to me in many ways because the only way w goes to zero is if Vo or R go to infinity or if Ѳ is initially zero which would just means the object is going straight towards the planet. Also notice, if Vo is zero the object will just approach the planet at Ѳ which also makes sense.

Let me know what you guys think. Thanks.

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