The International Space Station (ISS) orbits Earth at an altitude of 4.0 x 10^5 m above the surface of the planet. The radius of the Earth is 6.37 x 10^6 m. At what velocity must the ISS be moving in order to stay in its orbit?
a. 7.66 x 10^3 m/s
b. 7.91 x 10^3 m/s
c. 8.17 x 10^3 m/s
d. 3.12 x 10^3 m/s
To stay in its orbit, the ISS has to maintain a balance between the centripetal force and the gravitational force.
The centripetal force is given by Fc = mv^2/r, where m is the mass of the ISS required to maintain at the orbit.
The gravitational force is given by Fg = G(m*Me)/r^2, where G is the gravitational constant, Me is the mass of the Earth, and r is the distance between the ISS and the center of the Earth.
Setting the centripetal force equal to the gravitational force:
mv^2/r = G(m*Me)/r^2
Mass cancels out:
v^2/r = G(Me)/r^2
Simplifying:
v^2 = G(Me)/r
v = √(G(Me)/r)
Substituting the given values:
v = √((6.67430 x 10^-11 N*m^2/kg^2)(5.972 x 10^24 kg))/(6.37 x 10^6 m)
Calculating the value:
v = √(3.98249174 x 10^14 N*m^2/kg^2)/(6.37 x 10^6 m)
v = √(6.23254767 x 10^7 m^2/s^2)/2.54 x 10^6 m
v = √(6.23254767 m^2/s^2)/2.54 m
v ≈ √(2.4524442197) m/s
v ≈ 1.564917916 m/s
Rounded to three significant figures:
v ≈ 1.56 x 10^3 m/s
Therefore, the velocity at which the ISS must be moving to stay in its orbit is 1.56 x 10^3 m/s, which is closest to option d. 3.12 x 10^3 m/s.