Now, let us suppose an object of mass small m is projected straight up from the Earth, whose mass we this time call capital M. Can we find the speed with which it must be projected to escape the gravitational pull of the Earth? The total mechanical energy of the rocket at any point is given by the energy equation that we used earlier. On the surface of the Earth, r is simply the radius of Earth r-E, and let us call the initial velocity of the object on Earth v sub i. In this case, the total mechanical energy is one half m-v-i squared minus the the gravitational constant G times the product of the masses of the Earth and the rocket divided by r-e. At the maximum altitude, the velocity of the rocket is zero, and we'll call its distance from the center of the Earth r-max. The total energy reduces to negative G times the product of the mass of the Earth and the mass of the rocket, divided by the maximum altitude r-max. Since total mechanical energy is conserved, the total energy of the projectile on Earth should equal its total mechanical energy at the maximum altitude. By setting these two values equal, and simplifying. The initial speed to escape the gravitational pull from Earth is given by: The minimal escape speed is then obtained by setting r max to infinity. This is speed for which the projectile will reach infinity with a final speed of zero. Therefore the initial velocity is equal to the square root of two G times the mass of the Earth divided by the radius of the Earth. we obtain that the initial velocity is equal to the square root of two G times the mass of the Earth divided by the radius of the Earth. Let's do an example. Calculate the escape velocity of a 3000-kilogram rocket from the surface of the Earth. The mass of the Earth is equal to 5.98 times 10 to the 24th kilograms, and the radius of the Earth is equal to 6.37 times 10 to the 6th meters. ... Notice that the result of 11.2 kilometers per second obtained does not depend on the projectile's mass. Thus, this value is the value of the escape velocity of any object from the surface of the Earth.

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