"Calculating Work Needed to Change Orbits

[Identity]

Homework Statement

A shuttle orbiting the Earth at 400km deploys a satellite of mass 800kg into orbit a further 200km from earth. Calculate the work that must be done by the shuttle to deploy the sattelite.

Homework Equations

$$E_{k} = \frac{GMm}{2r}$$, $$U = -\frac{GMm}{r}$$

The Attempt at a Solution

I tried using $$W = \Delta E_{k}$$ to solve the problem:

$$800GM\left(\frac{1}{2 \times 600000} - \frac{1}{2 \times 400000}\right)=-1.334 \times 10^{11} J$$

But the solutions gives $$1.4 \times 10^9 J$$

Moreover, using the following method gives a different answer:

$$\int_{400000}^{600000} \frac{GM(800)}{r^2}dr=2.658 \times 10^{11} J$$

But I thought $$E_{tot} = E_{k} + U$$, so $$\Delta E_{k} = -\Delta U$$ ??

Could someone tell me why my methods do not work and what the correct method is for dealing with this? Thankyou

 

[tiny-tim]

a case of mistaken Identity …

Hi Identity!

A shuttle orbiting the Earth at 400km deploys a satellite of mass 800kg into orbit a further 200km from earth. Calculate the work that must be done by the shuttle to deploy the sattelite.
…
Could someone tell me why my methods do not work

They do work …

but you need to measure r from … ? ​

 

[Identity]

Oh right I need to measure r from the centre of mass XD thanks tiny-tim

But

$$800GM\left(\frac{1}{2(6.4 \times 10^6 + 600000)} - \frac{1}{2(6.4 \times 10^6 + 400000)}\right)=-6.7 \times 10^8 J$$

which is still different!

Also, when the satellite moves from 400km to 600km, is the total energy of the satellite constant? If so, why do I get different answers when working with GPE as opposed to Kinetic energy?

 

[tiny-tim]

$$800GM\left(\frac{1}{2(6.4 \times 10^6 + 600000)} - \frac{1}{2(6.4 \times 10^6 + 400000)}\right)=-6.7 \times 10^8 J$$

which is still different!

WIhtout seeing the details of your calculation, I can't check it …

but it would be a lot easier to use the formula 1/r - 1/(r + ∆) ~ ∆/r²

anyway, going to bed now … :zzz:

 

[Identity]

lol I just plugged the whole calculation into my calculator...

Sorry I'm having some trouble understanding. Can you show me how you would do it?