Question

A less-than-successful inventor wants to launch small satellites into orbit by launching them straight up from the surface of the earth at very high speed. a. With what speed should he launch the satellite if it is to have a speed of 500 m/s at a height of 400 km? ignore the air resistance.

Answer 1

Answer:

2763.53411 m/s

Explanation:

M = Mass of Earth = 5.972 × 10²⁴ kg

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

r = Radius of Earth = 6371000 m

$v_i$ = Launch velocity

$v_f$ = Final velocity = 500 m/s

h = Altitude = 400 km

m = Mass of satellite

As the energy of the system is conserved we have

$U_i+K_i=U_f+K_f\\\Rightarrow -\dfrac{GMm}{r}+\dfrac{1}{2}mv_i^2=-\dfrac{GMm}{r+h}+\dfrac{1}{2}mv_f^2\\\Rightarrow -\dfrac{GM}{r}+\dfrac{1}{2}v_i^2=-\dfrac{GM}{r+h}+\dfrac{1}{2}v_f^2\\\Rightarrow \dfrac{1}{2}v_i^2=\dfrac{GM}{r}-\dfrac{GM}{r+h}+\dfrac{1}{2}v_f^2\\\Rightarrow v_i=\sqrt{2GM(\dfrac{h}{r(r+h)})+v_f^2}\\\Rightarrow v_i=\sqrt{2\times 6.67\times 10^{-11}\times 5.972\times 10^{24}(\dfrac{400000}{6.371\times 10^6(6.371\times 10^6+400000)})+500^2}\\\Rightarrow v_i=2763.53411\ m/s$

The launch speed of the satellite should be 2763.53411 m/s