Respuesta :
Answer:
[tex]v_A=7667m/s\\\\v_B=7487m/s[/tex]
Explanation:
The gravitational force exerted on the satellites is given by the Newton's Law of Universal Gravitation:
[tex]F_g=\frac{GMm}{R^{2} }[/tex]
Where M is the mass of the earth, m is the mass of a satellite, R the radius of its orbit and G is the gravitational constant.
Also, we know that the centripetal force of an object describing a circular motion is given by:
[tex]F_c=m\frac{v^{2}}{R}[/tex]
Where m is the mass of the object, v is its speed and R is its distance to the center of the circle.
Then, since the gravitational force is the centripetal force in this case, we can equalize the two expressions and solve for v:
[tex]\frac{GMm}{R^2}=m\frac{v^2}{R}\\ \\\implies v=\sqrt{\frac{GM}{R}}[/tex]
Finally, we plug in the values for G (6.67*10^-11Nm^2/kg^2), M (5.97*10^24kg) and R for each satellite. Take in account that R is the radius of the orbit, not the distance to the planet's surface. So [tex]R_A=6774km=6.774*10^6m[/tex] and [tex]R_B=7103km=7.103*10^6m[/tex] (Since [tex]R_{earth}=6371km[/tex]). Then, we get:
[tex]v_A=\sqrt{\frac{(6.67*10^{-11}Nm^2/kg^2)(5.97*10^{24}kg)}{6.774*10^6m} }=7667m/s\\\\v_B=\sqrt{\frac{(6.67*10^{-11}Nm^2/kg^2)(5.97*10^{24}kg)}{7.103*10^6m} }=7487m/s[/tex]
In words, the orbital speed for satellite A is 7667m/s (a) and for satellite B is 7487m/s (b).
