Universal Gravitation and spacecraft

[mybrohshi5]

Homework Statement

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.

Homework Equations

k = 1/2 mv²

U = -GMm/R_e

The Attempt at a Solution

So i found this question fine by doing

k_i + U_i = k_f + U_f

k_i = 0

-GMm/(r + R_e) = 1/2mv² + -GMm/R_e

reduce to get

-GM/infinty = 1/2v² + -GM/R_e

0 = 1/2v² + -62413636.36

v = 11172 m/s



The next part is what i don't get

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

so r = 7.337 x 10⁷

Using the same equation

-GM/(r) = 1/2v² + -GM/R_e

-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 don't get. The U_i should not be zero it should be -GM/(r) and the Uf should be -GM/(R_e) not -GM/(r)

I don't get why the way i did it on this part where i got v = 10675 doesn't give me the right answer of 3294 :(

 

[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)?

 

[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 :(