A spacecraft whose mass is 1.5x10^5 kg is placed in orbit 150 miles above the surface of the Earth. (a)Determine the minimum energy required to achieve this orbit if the spacecraft is launched from rest from the surface of the Earth. (b)Determine the minimum energy required to put this spacecraft in this orbit if it were launched from a plane flying at 300mph at an altitude of 34,000ft above the Earth's surface.

Question

snowsurf · Answer

For part a.)  as long there are no other forces then KE + PE = 0

$$KE = \frac{ GM_1M_2 }{ r }$$

Where r is the radius of the Earth and M are the masses of Earth and the object in question.

part b is bit different because your above the Earth surface and traveling in a plane at 300 mph. You need to subtract the plane altitude from the height of the orbit which is 150 miles. Once you find that difference that will be your r. Note make sure you convert these units.

anonymous · Answer

A) Answer is -9.066x10^12joules

irishboy123 · Answer

at rest on the earth surface the satellite has potential energy $U_e = -\dfrac{GMm}{r_e}$

in orbit will have $U_o = -\dfrac{GMm}{r_o}$. the negative sign ensures that U increases with distance from earth, we are only interested in changes in U so the sign itself does not matter....

next the satellite will require an orbit speed to stay in orbit which you can get from $\dfrac{mv^2}{r} = \dfrac{GMm}{r^2}$ so that  kinetic energy $T_o = \dfrac{mv^2}{2} = \dfrac{GMm}{2r}$

the energy required is therefore \E = (T_o + (U_o - U_e)\)

strictly speaking you should include the kinetic energy the satellite already has by virtue if its being fixed to the earth's  rotation but i suspect that is a small number in the context

for b) repeat the process...

irishboy123 · Answer

oh dear, latex glitch :p

 $E = T_o + (U_o - U_e)$