A satellite revolves in the geostationary orbit but in a direction east to west. The time interval between its successive passing about a point on the equator is:

Options

A. 4h
B. 6h
C. 12h
D. 24h

Answer

To solve the problem of finding the time interval between successive passes of a satellite in a geostationary orbit moving from east to west, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Geostationary Orbit**:
A geostationary satellite has a time period of revolution that is equal to the rotational period of the Earth. This period is 24 hours (or 1 day).

**Hint**: Remember that geostationary satellites remain fixed over a point on the equator.

2. **Calculate Angular Velocity**:
The angular velocity (ω) of the satellite can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period. For a geostationary satellite, \( T = 24 \) hours = 86400 seconds.
\[ \omega = \frac{2\pi}{86400} \text{ rad/s} \approx \frac{\pi}{43200} \text{ rad/s} \]

**Hint**: Angular velocity is the rate of change of angle per unit time.

3. **Angular Velocity of Earth**:
The angular velocity of the Earth (ω_e) is the same as that of the satellite because it is geostationary:
\[ \omega_e = \frac{2\pi}{T} = \frac{2\pi}{86400} \text{ rad/s} \approx \frac{\pi}{43200} \text{ rad/s} \]

**Hint**: Both the satellite and Earth have the same angular velocity in a geostationary orbit.

4. **Relative Angular Velocity**:
Since the satellite is moving in the opposite direction (east to west), we need to consider the relative angular velocity (ω_r):
\[ \omega_r = \omega_e + \omega_s = \frac{\pi}{43200} + \frac{\pi}{43200} = \frac{2\pi}{43200} = \frac{\pi}{21600} \text{ rad/s} \]

**Hint**: When two objects move in opposite directions, their angular velocities add up.

5. **Calculate the Time Interval**:
The time interval (T_r) between successive passes can be calculated using:
\[ T_r = \frac{2\pi}{\omega_r} \]
Substituting the value of ω_r:
\[ T_r = \frac{2\pi}{\frac{\pi}{21600}} = 2 \times 21600 = 43200 \text{ seconds} \]
Converting seconds to hours:
\[ T_r = \frac{43200}{3600} = 12 \text{ hours} \]

**Hint**: To convert seconds to hours, divide by 3600.

### Final Answer:
The time interval between successive passes of the satellite about a point on the equator is **12 hours**.