Does anyone out there know an estimate of the minimum energy needed to launch a satellite into space? Or at least where I can find out? I'm desperate to find out, my physics assignment needs to be handed in soon...
I believe that it takes twice the amount of kinetic energy to launch a satellite as it takes to maintain the orbit of the satellite. The escape velocity is: v = sqrt(2GM/r)
From Newton's law of gravitation... a(r) = -gR² / r² Where R is the radius of the Earth and r is the distance of the satellite from the center of the Earth. Using the Chain Rule... a = dv/dt = dv/dr * dr/dt But dr/dt = v so... a = dv/dr * v Substitute this... dv/dr * v = -gR² / r² Separate the variables... vdv =-gR² * dr/r² Integrate both sides... 1/2 v² = gR² / r + C (*) On the Earth's surface where you are launching your satellite you have r = R and v = v_0 (the initial velocity). So we have... 1/2 v² _0 = gR² / R + C 1/2 v² _0 = gR + C Where C = 1/2 v² _0 - gR Substituting this solution from initial conditions back into the general solution (*) we have... 1/2 v² = gR² / r + 1/2 v² _0 - gR ___ v² = 2gR² / r + v² _0 - 2gR ___ If v² = 0 then v = 0 and the satellite will fall back to Earth. If v² = 2gR then v² = 2gR² / r + 0 and this velocity is always positive. Thus we need a velocity of at least v = √(2gR) to escape the pull of Earth's gravity. v
Yes, but that is the escape velocity which is something like 11 km/s. If you just want to get to a low orbit you need a speed of about 7 km/s, don't remember the altitude. The kinetic energy is as always mv²/2. Much cheaper!
True, but that orbit will decay rapidly back into Earth. I think the question is a little vague though. If the point of the launch was to simply get something into space and not keep it there then Omnignost's low orbit technique would suffice (not enough time to work it out ATM). However if the point was to get into space and onwards (like to the moon or something) then the minimum speed would have to be the escape velocity which is 11170km/s. So our question is: which one is it sandgroper?
If the object is to place the satellite into stable LEO you don't need to achieve escape velocity. To do this you do not apply all of your delta v at launch. First you launch into an orbit with an apogee of an altitude of 300km (and a perigee at launch point) this takes a delta v of about 8 km/sec. Upon reaching apogee you apply another 100 km/sec or so which raises your perigee up to 300km altitude. This puts you into a stable LEO, all for a total delta v of 8.1 km/sec. This works out to about 32,800,000 joules per Kg A more direct way of getting the minimum energy is by the formula: E = GMm(1/r-1/(2a+2r)) where G = gravitational constant M = mass of Earth m = mass of satellite r = radius of Earth a= altitude of orbit This formula doesn't take the rotation of the Earth into account, however, so to be more accurate you should add the term -107648cos²(L)m Where L is the latitude of the launch point. this gives E = GMm(1/r-1/(2a+2r)) -107648cos²(L)m
Or you could just look up the orbital formula for an elliptical orbit around the earth and figure out what energy you want for given radii and velocities. It should be pretty standard with derivation and such in any mechanics textbook, and I would recommend you look through it since central force motion is so useful.
Cheers, fellas Thanks a lot guys, you're real lifesavers... But I think my Year 12 Physics teacher will get suspicious if I start factoring in things like rotation of the Earth...
Sorry about the vagueness, oxymoron, but that is the actual wording of one of the questions I received. My teacher is the one being vague...
