Answer:
[tex]6.09294\times 10^{24}\ kg[/tex]
Explanation:
K = Kinetic energy
[tex]v_p[/tex] = Perigee speed = 4280 m/s
[tex]v_a[/tex] = Apogee speed = 3990 m/s
[tex]r_p[/tex] = Perigee Distance = 22500000 m
[tex]r_a[/tex] = Apogee Distance = 24100000 m
G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²
M = Mass of Earth
m = Mass of satellite
In this system the kinetic and potential energies are conserved
[tex]K_p+P_p=K_a+P_a\\\Rightarrow \frac{1}{2}mv_p^2-\frac{GMm}{r_p}=\frac{1}{2}mv_a^2-\frac{GMm}{r_a}\\\Rightarrow \frac{1}{2}m(v_p^2-v_a^2)+GMm\left(\frac{1}{r_a}-\frac{1}{r_p}\right)=0\\\Rightarrow M=\frac{v_a^2-v_p^2}{2G}\times \left(\frac{1}{r_a}-\frac{1}{r_p}\right)^{-1}\\\Rightarrow M=\frac{3990^2-4280^2}{2\times 6.67\times 10^{-11}}\times \left(\frac{1}{24100000}-\frac{1}{22500000}\right)^{-1}\\\Rightarrow M=6.09294\times 10^{24}\ kg[/tex]
The mass of the Earth is [tex]6.09294\times 10^{24}\ kg[/tex]
