The minimum kinetic energy needed for escape the Earth’s gravitation.

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Chapter 21, Problem 72AP

(a)

To determine

The minimum kinetic energy needed for escape the Earth’s gravitation.

(b)

To determine

The temperature for the minimum escape kinetic energy.

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The escape velocity from a massive object is the speed needed to reach an infinite distance from it and have just slowed to a stop, that is, to have just enough kinetic energy to climb out of the gravitational potential well and have none left. You can find the escape velocity by equating the total kinetic and gravitational potential energy to zero E = = muesc - GmM/r=0 Vesc = √2GM/r where G is Newton's constant of gravitation, M is the mass of the object from which the escape is happening, and r is its radius. This is physics you have seen in the first part of the course, and you should be able to use it to find an escape velocity from any planet or satellite. For the Earth, for example the escape velocity is about 11.2 km/s, and for the Moon it is 2.38 km/s. A very important point about escape velocity: it does not depend on what is escaping. A spaceship or a molecule must have this velocity or more away from the center of the planet to be free of its gravity, 1. In the atmosphere of…

(a) State Newton's law of gravitation in word form. (b) Use the law to show that that the escape velocity for a planet is given by the expression v = 2go R. You can assume that the gravitational potential per unit -GM where mass U, at the surface of the planet, is given by the expression U = R M and R are the mass and radius of the planet. (c) Assuming that the Earth is a uniform sphere of radius 6.4 x 10 m and mass 6.0 x 1024 kg, calculate: (i) the gravitational potential at the surface of the Earth and (ii) the gravitational potential at a point 6.0 x 105 m above the surface of the Earth. (d) A 5 kg mass is to be delivered to a point in space where the gravitational effect on the mass is negligible. Calculate: (i) the escape velocity for the mass. (iii) the work done in taking the mass to a point 6.0 x 105 m above the surface of the Earth. (iii) the work needed to deliver the mass to the point where the gravitational potential is zero.

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