First slide

Motion of planets and satellites

Question

A satellite moves eastwards very near the surface of the Earth in equatorial plane with speed (v₀). Another satellite moves at the same height with the same speed in the equatorial plane but westwards. If R = radius of the Earth and $ω$ be its angular speed of the Earth about its own axis. Then find the approximate difference in the two time period as observed on the Earth.

Moderate

A

$\frac{4 {πωR}^{2}}{v_{0}^{2} + R^{2} ω^{2}}$
B

$\frac{2 {πωR}^{2}}{v_{0}^{2} - R^{2} ω^{2}}$
C

$\frac{4 {πωR}^{2}}{v_{0}^{2} - R^{2} ω^{2}}$
D

$\frac{2 {πωR}^{2}}{v_{0}^{2} + R^{2} ω^{2}}$

Solution

$\begin{array}{l} T_{west} = \frac{2 πR}{v_{0} + Rω} and T_{east} = \frac{2 πR}{v_{0} - Rω} \Rightarrow \\ \Rightarrow ∆ T = T_{east} - T_{west} = \frac{2 π R}{V_{0} - R ω} - \frac{2 π R}{V_{0} + R ω} = [\frac{2 πR \times 2 R ω}{v_{0}^{2} - R^{2} ω^{2}}] = \frac{4 {πωR}^{2}}{v_{0}^{2} - R^{2} ω^{2}} \end{array}$

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