Question

A satellite orbits the Earth in an elliptical orbit. At perigee its distance from the center of the Earth is 22500 km and its speed is 4280 m/s. At apogee its distance from the center of the Earth is 24100 km and its speed is 3990 m/s.using this information calculate the mass of the earth. please explain how you arrived to your answer. I have an exam coming up.

Answer 1

Answer:

$6.09294\times 10^{24}\ kg$

Explanation:

K = Kinetic energy

$v_p$ = Perigee speed = 4280 m/s

$v_a$ = Apogee speed = 3990 m/s

$r_p$ = Perigee Distance = 22500000 m

$r_a$ = 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

$K_p+P_p=K_a+P_a\\\Rightarrow \frac{1}{2}mv_p^2-\frac{GMm}{r_p}=\frac{1}{2}mv_a^2-\frac{GMm}{r_a}\\\Rightarrow \frac{1}{2}m(v_p^2-v_a^2)+GMm\left(\frac{1}{r_a}-\frac{1}{r_p}\right)=0\\\Rightarrow M=\frac{v_a^2-v_p^2}{2G}\times \left(\frac{1}{r_a}-\frac{1}{r_p}\right)^{-1}\\\Rightarrow M=\frac{3990^2-4280^2}{2\times 6.67\times 10^{-11}}\times \left(\frac{1}{24100000}-\frac{1}{22500000}\right)^{-1}\\\Rightarrow M=6.09294\times 10^{24}\ kg$

The mass of the Earth is $6.09294\times 10^{24}\ kg$