A projectile is shot directly away from Earth's surface. Neglect the rotation of the Earth. What multiple of Earth's radius RE gives the radial distance (from the Earth's center) the projectile reaches if (a) its initial speed is 0.241 of the escape speed from Earth and (b) its initial kinetic energy is 0.241 of the kinetic energy required to escape Earth. (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

(a) r = 1.062·R =

(b) r =

(c) Zero

Explanation:

Here we have escape velocity v given by

and the maximum height given by

Therefore, when the initial speed is 0.241v we have

v = so that;

v² =

v² =

is then

Which gives

or

r = 1.062·R

(b) Here we have

Therefore we put   in the maximum height equation to get

From which we get

r = 1.32·R

(c) The we have the least initial mechanical energy, ME given by

ME = KE - PE

Where the KE = PE required to leave the earth we have

ME = KE - KE = 0

The least initial mechanical energy to leave the earth is zero.

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