Find r'' and Theta''for the Spaceship

[Northbysouth]

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''

ƩF^r = ma_r = m(r'' - rθ'²) = -Gmm_E/r²

r'' = -Gm_E/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''

ma_θ = 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.

 

[tms]

For r''

m(r'' - rθ'²) = -Gmm_E/r²

Haven't you left out something on the left hand side?

 

[Northbysouth]

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.

 

[tms]

Yes, but the left hand side is missing something.

 

[haruspex]

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

r'' = -Gm_E/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?