A projectile is shot directly away from Earth\'s surface. Neglect the rotation o

Question

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 0.560 of the escape speed from Earth and (b) its initial kinetic energy is 0.560 of the kinetic energy required to escape Earth? (Give your answers as unitless numbers.) (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

Explanation / Answer

Given Data

v is escape velocity

R is radius of earth

Formula

        v= root(2GM/R)

        v^2=2GM/R

      1/2v^2 =GM/R

(a) inital speed =0560v

       1/2m(v^2-u^2) =GMm/r

       1/2v^2(1-(0.560)2 =GM/r

       1/2v^2 (0.06864) =GM/R

Replacing 1/2v^2 by GM/R and cancelling on both sides

       0.6864/R =1/r

      r/R =1.4568

(b) Given 1/2mu^2 =0.560 of 1/2 mv^2

1/2 mv^2 - 1/2 mu^2 =GMm/r

(1-0.560) 1/2v^2 =GM/r

Replacing 1/2 v^2 by GM/R and cancelling GM both sides

(0.44)/R =1/r

r/R =2.2727

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