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What goes up must come down, unless…. Escape v elocity . Consider masses M and m placed in space a distance R from each other. The two masses have gravitational potential energy, which is stored in their gravitational field.
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What goes up must come down, unless… Escape velocity
Consider masses M and m placed in space a distance R from each other. The two masses have gravitational potential energy, which is stored in their gravitational field. • This energy is there because work had to be done to move one of the masses from infinity to the position near the mass. Gravitational Potential Energy
How fast does it have to go to not come back down? • The total energy of a mass m moving near a large stationary mass M is ; where v is the speed of m when it is a distance r from M. (if M is also free to move, then you need to include a term ; where u is the speed of M.) • The only force acting on m is the gravitational attraction of M. Suppose that m is launched with a speed v0 from M. Will m escape from the pull of M and move very far away from it? Total energy must be zero or positive Escape velocity
E>0: mass escapes and never returns • E<0: mass moves out a certain distance, but returns – trapped • E=0: the critical case separating the other two – mass barely escapes • Which of these would be good for a space shuttle? Escape velocity
How fast something must go in order to escape the gravitational pull of an object. • Smallest v for which v∞ = 0 • Kinetic energy = gravitational potential energy Not dependent on the mass of the object Does not account for frictional forces (air resistance) or other external forces. Escape velocity
What must the radius of a star of mass M be such that the escape velocity from the star is equal to the speed of light, c? Example – the Swarzchild Radius
Compute the swarzchild radius of the earth and the sun. Example
