Answer to Question #266929 in Mechanics | Relativity for Jay ar

Question

A projectile is fired straight upward from the Earth's surface at the south pole with an initial speed equal to one third the escape speed. Ignoring air resistance, determine how far from the center of the earth the projectile travels before stopping momentarily.

Answer 1

The radius of the Earth is 6.38 10⁶ m

G = Gravitational constant

m = Mass of projectile

M = Mass of Earth

R = Radius of Earth

h = Altitude

r = Distance from the center of Earth

v_e = Escape velocity

Initial velocity

$u = \frac{1}{3}v_e \\ u = \frac{1}{3} \sqrt{\frac{2GM}{R}}$

Kinetic energy at the surface

$K = \frac{1}{2}mu^2 \\ K = \frac{1}{2} m (\frac{1}{3} \sqrt{\frac{2GM}{R}})^2 \\ K = \frac{1}{9}m \frac{GM}{R}$

Potential + Kinetic energy at the surface = Potential energy at the max height

$-\frac{GMm}{R} + \frac{1}{9}m \frac{GM}{R} = \frac{GMm}{r}$

Cancelling G, M, and m

$-\frac{1}{R} + \frac{1}{9} \frac{1}{R} = -\frac{1}{r} \\ \frac{-9+1}{9R} = -\frac{1}{r} \\ \frac{8}{9R} = \frac{1}{r} \\ r = \frac{9}{8}R \\ r = \frac{9}{8} \times 6.38 \times 10^6 \\ r = 7177500 \;m$