To solve the problem of finding the ratio of the kinetic energy of satellite B to that of satellite A in the reference frame fixed to the Earth, we can follow these steps:
### Step 1: Understand the Motion of the Satellites
- Satellite A moves in the same direction as the Earth's rotation, while satellite B moves in the opposite direction.
- Both satellites are in the same orbit, hence they have the same radius (r) from the center of the Earth.
### Step 2: Calculate the Angular Velocity of the Satellites
- The angular velocity (ω) for both satellites can be derived from the formula:
\[
\omega = \sqrt{\frac{g}{r}}
\]
- Given \( g = 9.8 \, \text{m/s}^2 \) and the radius of the Earth \( r = 6.37 \times 10^6 \, \text{m} \), we can calculate ω.
### Step 3: Find the Angular Velocity with Respect to the Earth
- The angular velocity of the Earth (ω_e) can be calculated using:
\[
\omega_e = \frac{2\pi}{T}
\]
where \( T = 24 \, \text{hours} = 86400 \, \text{seconds} \).
- Thus,
\[
\omega_e = \frac{2\pi}{86400} \approx 7.3 \times 10^{-5} \, \text{rad/s}
\]
### Step 4: Calculate the Angular Velocities of Satellites A and B
- For satellite A (moving in the same direction as Earth's rotation):
\[
\omega_{A,E} = \omega_A - \omega_e
\]
- For satellite B (moving in the opposite direction):
\[
\omega_{B,E} = \omega_B + \omega_e
\]
### Step 5: Calculate the Kinetic Energy Ratio
- The kinetic energy (KE) of a satellite is given by:
\[
KE = \frac{1}{2} m v^2
\]
- The velocities can be expressed in terms of angular velocities:
\[
v_A = \omega_{A,E} \cdot r \quad \text{and} \quad v_B = \omega_{B,E} \cdot r
\]
- Thus, the ratio of kinetic energies becomes:
\[
\frac{KE_B}{KE_A} = \frac{v_B^2}{v_A^2} = \frac{(\omega_{B,E} \cdot r)^2}{(\omega_{A,E} \cdot r)^2} = \frac{\omega_{B,E}^2}{\omega_{A,E}^2}
\]
### Step 6: Substitute Values and Calculate
- Substitute the calculated angular velocities into the ratio:
\[
\frac{KE_B}{KE_A} = \frac{(131.3 \times 10^{-5})^2}{(116.7 \times 10^{-5})^2}
\]
- Simplifying gives:
\[
\frac{KE_B}{KE_A} \approx 1.27
\]
### Final Answer
The ratio of the kinetic energy of satellite B to that of satellite A is approximately **1.27**.
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