# Universal Gravitation and spacecraft

1. Apr 21, 2010

### mybrohshi5

1. The problem statement, all variables and given/known data

Very far from earth (r = infinity), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is M_e=5.97×1024 kg and its radius is R_e=6.38×106 m.
G = 6.67 x 10-11

Find the speed V of the spacecraft when it eventually crashes into the earth.

2. Relevant equations

k = 1/2 mv2

U = -GMm/Re

3. The attempt at a solution

So i found this question fine by doing

ki + Ui = kf + Uf

ki = 0

-GMm/(r + Re) = 1/2mv2 + -GMm/Re

reduce to get

-GM/infinty = 1/2v2 + -GM/Re

0 = 1/2v2 + -62413636.36

v = 11172 m/s

The next part is what i dont get

Now find the spacecraft's speed when its distance from the center of the earth is r=x(Re), where x=11.5

so r = 7.337 x 107

Using the same equation

-GM/(r) = 1/2v2 + -GM/Re

-5427272.7 = 1/2v2 + -62413636.36

v = 10675 m/s

but that is wrong.

The right answer is 3294 m/s

to get 3294 m/s you have to do

0 = 1/2v2 + -GM/(r) ...........r = 7.337 x 107

This step right here is what i dont get. The Ui should not be zero it should be -GM/(r) and the Uf should be -GM/(Re) not -GM/(r)

I dont get why the way i did it on this part where i got v = 10675 doesnt give me the right answer of 3294 :(

Last edited: Apr 21, 2010
2. Apr 21, 2010

### zachzach

The second part is asking the same as the first part except instead of wanting v when it crashes to Earth, it wants v at xR(E). It is till starting from r=infinity. Why would Uf = -GMm/R(E) when it is not located at R(E)?

3. Apr 21, 2010

### mybrohshi5

Oh ok i got it. Thanks. I thought it still wanted it when crashing into earth from the new r. Mastering Physics is always so wordy :(

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