Find r'' and Theta''for the Spaceship

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Homework Help Overview

The problem involves determining the second derivatives of the radial and angular components of motion for a spacecraft in an elliptical orbit. The spacecraft's speed and gravitational parameters are provided, and the original poster is attempting to calculate r'' and θ'' using these values.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to apply Newton's second law in polar coordinates to find r'' and θ''. They express confusion over their calculations and seek clarification on their approach.
  • Some participants question whether all forces acting on the spacecraft have been considered in the equations provided.
  • Others suggest that there may be a missing term in the force balance equation used for r''.

Discussion Status

The discussion is ongoing, with participants exploring the original poster's calculations and questioning the completeness of the force equations. There is no explicit consensus yet, but some guidance has been offered regarding potential oversights in the equations.

Contextual Notes

The original poster has provided specific values for gravitational acceleration and the radius of the Earth, which may influence the calculations. There is also mention of a diagram that has not been shared in the text, which could be relevant to the problem setup.

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Homework Statement


The spacecraft P is in the elliptical orbit shown. At the instant represented, its speed is v = 13164 ft/sec. Determine the corresponding values of and . Use g = 32.23 ft/sec2 as the acceleration of gravity on the surface of the Earth and R = 3959 mi as the radius of the earth.

I have uploaded an image of the solution.

Homework Equations





The Attempt at a Solution



r = 16388 miles

I've found r' = 9050.85 ft/s and θ' = 0.00011047 rad/s

But I cannot for the life of mt figure out why I can get θ'' or r''.

For r''

ƩFr = mar = m(r'' - rθ'2) = -GmmE/r2

r'' = -GmE/r2 + rθ'2
= -3.439x10-8ft4/lbfs4+ 4.095x1023 lbf=s2/lbf)/ 16388miles*5280) + (16388miles*5280)*0.00010472

r'' = -.93236 ft/s2

It says this is wrong and I've at a loss for where my mistake is.

Likewise for θ''

maθ = m(rθ''+2r'θ') = 0

Mass cannot equal zero, therefore the summation in the brackets must be equal to zero

θ'' = -2r'θ/r

= _2*9050ft/s*(133.63°(π/180))

θ'' = -.0004879

This is also incorrect.

Any help would be appreciated. Thanks.
 

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Northbysouth said:
For r''

m(r'' - rθ'2) = -GmmE/r2
Haven't you left out something on the left hand side?
 
Last edited:
I'm sorry, but I don't see it. As far as I can tell the only force acting on the object is the force of the gravitational pull from the Earth.
 
Yes, but the left hand side is missing something.
 
Northbysouth said:
I've found r' = 9050.85 ft/s and θ' = 0.00011047 rad/s

r'' = -GmE/r2 + rθ'2
= -3.439x10-8ft4/lbfs4+ 4.095x1023 lbf=s2/lbf)/ 16388miles*5280) + (16388miles*5280)*0.00010472
You dropped a 1. Is that just a typo in the OP?