Launching Satellites and Escape Energy (this is rocket science baby!) Recall: mg = GMm / r2 Recall: the units of g = N / kg N m2 X kg = N kg2 1 kg m2

1. Sketch Force on 1kg Object Distance From Center of Earth (in Earth Radii) Earth 5 10 Equation of the curve: ∞ 0 gpE Fg= mg 9.8 / 16 9.8 / 25 9.8 / 9 9.8 9.8 / 4 NOTE: The x-axis is the distance from the center of the Earth cumulative AREA UNDER THE CURVE = N x m = J = GPE that needs to be overcome!!!! A changing g field – unlike Newtonian gPE = mgh Exponential decay of gPE with increased distance!

The conservation of Energy Relate the KE to the gPE Final d = ∞ gPE at ∞ = 0 final v = 0 So final total E = 0 As the object moves from the planet, the KE lost = gPE overcome In other words, the KE + gPE = 0 ½ mv2 = - GMm/r

What is the Escape Energy from 3 Earth Radii for a 1 kg object. ( What is the Escape Energy from 3 Earth Radii for a 1 kg object??? (*****if the object has no horizontal velocity) Shade the area which represents the escape energy from 3 Earth radii. ? the expression for this escape energy. ? amount of energy for the 1kg object to escape from 3 Earth radii.

Approach the pbm. Like this: GMEm - GMm = the expression 3r ∞ 6.67 x 10-11 (5.97 x 1024) 1 = 2.1x107 J = E 1.89 x 107

? V Required to escape from the surface of the Earth (start with escape energy equation first) Shade the area that is required to move the object from the surface of the Earth to an infinite distance. GMm - GMm 1r ∞ ½ mv2 = GMEm / r v = 2GME / rE v = 1.1 x 104 m/s

4. Escape Speed Find the escape speed from the surface of Jupiter please. Did you calculate 6 x 104 m/s

Find the expression for the kinetic energy of a satellite of mass m in a circular orbit of radius r around Earth Hint: relate Fc to Fg to solve for v first. Fc = Fg mv2 = GMm so v = GM r r2 r KE = ½ mv2 = ½ m GM sub in for v2 r

NOW you have the basics down it’s time for the fun stuff

5. Find the expression and value (in J) for the launch energy required to launch a 1000kg satellite into a 300km orbit. NOTE: There are two parts: lift up the satellite and provide the kinetic energy to keep it up.

Get it up GMm - GMm rE rE + 300,000 m = 2.8 x 109 J

Keep it up Requires KE (horizontal v) Fc = Fg mv2 = GMm so v2 = GM r r2 r KE = ½ mv2 = ½ m GM r

KE = ½ mv2 = ½ m GM rE + 300,000m = 2.98 x 1010 J

To summarize the total expression GMm - GMm ½ GMm rE rE + 300,000 m + rE + 300,000m = 3.26 x 1010 J KNOW HOW TO DO THIS !!!!!!

ONE LAST ONE Find the expression for the energy required to escape from this orbit (considering it is already moving). HINT: It takes less energy to escape from orbit than from the same point not in orbit. GMEm - GMm - ½ m GM rorbit ∞ rorbit

Because the object is already in orbit, we would: (gPE at 300km above surface – gPE at infinity) – KE that keeps it in orbit. If you do the calculation for v, you should find a value around 7750 m/s

WELL DONE!!!!