Answer:
7177500 m
797500 m
Explanation:
G = Gravitational constant
m = Mass of projectile
M = Mass of Earth
R = Radius of Earth
h = Altitude
r = Distance from the center of Earth
[tex]v_e[/tex] = Escape velocity
Initial velocity
[tex]u=\frac{1}{3}v_e\\\Rightarrow u=\frac{1}{3}\sqrt{\frac{2GM}{R}}[/tex]
Kinetic energy at the surface
[tex]K=\frac{1}{2}mu^2\\\Rightarrow K=\frac{1}{2}m\left(\frac{1}{3}\sqrt{\frac{2GM}{R}}\right)^2\\\Rightarrow K=\frac{1}{9}m\frac{GM}{R}[/tex]
Potential + Kinetic energy at the surface = Potential energy at the max height
[tex]-\frac{GMm}{R}+\frac{1}{9}m\frac{GM}{R}=\frac{GMm}{r}[/tex]
Cancelling G, M, and m
[tex]-\frac{1}{R}+\frac{1}{9}\frac{1}{R}=-\frac{1}{r}\\\Rightarrow \frac{-9+1}{9R}=-\frac{1}{r}\\\Rightarrow \frac{8}{9R}=\frac{1}{r}\\\Rightarrow r=\frac{9}{8}R\\\Rightarrow r=\frac{9}{8}\times 6.38\times 10^6\\\Rightarrow r=7177500\ m[/tex]
Distance of the max height from the center of earth is 7177500 m
[tex]R+h = r\\\Rightarrow h=r-R\\\Rightarrow h=7177500-6.38\times 10^6\\\Rightarrow h=797500\ m[/tex]
The altitude of the projectile is 797500 m
