Answer:
The height (h) will be: [tex]\frac{3}{4}R =h[/tex]
Explanation:
The scape speed equation is given by:
[tex]v_{scape}=\sqrt{\frac{2GM}{R}}[/tex]
Now, the speed of the missile is
[tex]v_{missile}=\frac{1}{2}v_{scape}[/tex]
[tex]v_{scape}=\frac{1}{2}\sqrt{\frac{2GM}{R}}[/tex]
Using the conservation of energy, we can find the maximu height of the missile.
[tex]E_{i}=E_{f}[/tex]
[tex]\frac{1}{2}mv_{scape}^{2}-mgR =-mgh[/tex]
[tex]\frac{1}{2}\frac{2GM}{4R}-gR =-gh[/tex]
[tex]\frac{GM}{4R}-gR =gh[/tex]
Let's recall that g = GM/R², using the equivalence principle. When R is the radius of the earth and M is the mass of the earth.
[tex]\frac{1}{4}gR-gR =-gh[/tex]
[tex]\frac{1}{4}R-R =-h[/tex]
Therefore the height (h) will be:
[tex]\frac{3}{4}R =h[/tex]
I hope it helps you!
