Correct option is (1) 1.33 × 103 s
Given: Mass of the satellite A = 200 kg
Mass of the satellite B = 400 kg
Height of satellite A = 600 Km
Height of the satellite B = 1600 Km
The radius of satellite rA = 6400 + 600
= 7000 Km
= 7000 × 103 m
and the radius of satellite rB = 6400 + 1600
= 8000 Km
= 8000 ×103 m
Now, by using equation (1) we have;
The time period of A, \({T_A}^2 = \frac{4\pi^2 {r_A}^3}{GM_e}\)
⇒ \({T_A} =\sqrt{ \frac{4\pi^2 {r_A}^3}{GM_e}}\)
Now, on putting the values we have;
\(T_A = \sqrt{\frac{4\pi^2(7000\times 10^{3})^3}{GM_e}}\)
and the time period of B, \({T_B}^2 = \frac{4\pi^2 {r_B}^3}{GM_e}\)
⇒ \({T_B} =\sqrt{ \frac{4\pi^2 {r_B}^3}{GM_e}}\)
Now, on putting the values we have;
\(T_B = \sqrt{\frac{4\pi^2(8000\times 10^{3})^3}{GM_e}}\)
Now,
TB - TA = \(\sqrt{\frac{4\pi^2(8000\times 10^{3})^3}{GM_e}} - \sqrt{\frac{4\pi^2(7000\times 10^{3})^3}{GM_e}} \)
⇒ TB - TA = \(2\pi\sqrt{\frac{(8000\times 10^{3})^3}{GM_e}} - \sqrt{\frac{(7000\times 10^{3})^3}{GM_e}} \)
⇒ TB - TA = \(2\pi\sqrt{\frac{(8000 \times 10^3)^3 - (7000 \times 10^3)^3}{GM_e}}\)
⇒ TB - TA = \(2\times 3.14\sqrt{\frac{(8000 \times 10^3)^3 - (7000 \times 10^3)^3}{6.674 \times 10^{-11} \times 6 \times 10^{24}}}\)
⇒ TB - TA = 1.33 × 103 s