Motion, Forces and Energy Gravitation Part 2 Initially we assumed that the gravitational force on a body is constant. But now we know that the acceleration due to gravity is a function of its height above the Earth’s surface: g = g(h) We can find a new expression for the gravitational potential energy by calculating the work done by gravity when the object’s position changes from r 1 to r 2 : Usually, we choose r 1 to be equal to infinity, so that generally we can write:

Total Potential Energy of a System of Particles m1m1 m2m2 m3m3 r 12 r 23 r 13 This U total represents the work needed to Separate all three particles by infinity.

Energy Considerations M r v m In an isolated system, Etotal is constant so:

Escape Speed V f =0 ViVi Mass, m r max KE + U at launch point U at distance r max from surface (KE=0) If we know v i, then we can calculate height h reached by the object. If r max tends to infinity, then: If v i >V esc, then the object escapes the Earth’s gravitational field and carries excess kinetic energy.

Escape Speed and Thermal Physics Note that the mass of the “escaping” object does not figure in the equation above for escape velocity. The escape velocity for Earth is 11.2 km/s. Speed (m/s) Number of Molecules, N V esc N2N2 H2H2 Heavy gases are more abundant than lighter ones on planets and why nearly all the hydrogen and helium has escaped from the Earth’s surface.

Escape speeds for the planets and the sun. Body Escape Speed (km/s) Mercury 4.3 Venus 10.3 Earth11.2 Moon2.3 Mars5.0 Jupiter60 Saturn36 Uranus22 Neptune24 Pluto1.1 Sun618