To determine the satellite's altitude, we can use the principles of gravitational force and centripetal force, which are in balance for a satellite in a stable circular orbit.

The gravitational force acting on the satellite is given by:

$F_g = \frac{G M m}{r^2}$

Where:

• $G$ is the universal gravitational constant, $G = 6.67259 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$,
• $M$ is the mass of the Earth, $M = 5.98 \times 10^{24} \, \text{kg}$,
• $m$ is the mass of the satellite (it cancels out in the final equation),
• $r$ is the distance from the center of the Earth to the satellite.

The centripetal force required to keep the satellite in orbit is given by:

$F_c = \frac{m v^2}{r}$

Where:

• $v = 6.4 \times 10^3 \, \text{m/s}$ is the orbital speed of the satellite.

Since the gravitational force provides the necessary centripetal force, we set the two forces equal to each other:

$\frac{G M m}{r^2} = \frac{m v^2}{r}$

The mass $m$ of the satellite cancels out:

$\frac{G M}{r^2} = \frac{v^2}{r}$

Multiplying both sides by $r$, we get:

$\frac{G M}{r} = v^2$

Solving for $r$:

$r = \frac{G M}{v^2}$

Now, let's plug in the values:

• $G = 6.67259 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$,
• $M = 5.98 \times 10^{24} \, \text{kg}$,
• $v = 6.4 \times 10^3 \, \text{m/s}$.

Once we find $r$, we can subtract the Earth's radius to find the altitude:

$\text{Altitude} = r - R_{\text{Earth}}$

Where $R_{\text{Earth}} = 6370 \, \text{km}$.

Let me calculate this for you.The satellite's altitude above the surface of the Earth is approximately $3371.72 \, \text{km}$.

Would you like more details or have any questions?

Here are five related questions:

1. How does the mass of the satellite affect its orbital altitude?
2. What would happen if the satellite's speed was higher than $6.4 \, \text{km/s}$?
3. How does the altitude change if the Earth's mass were different?
4. What other forces (e.g., atmospheric drag) can affect a satellite's orbit at low altitudes?
5. How can this calculation change for elliptical orbits?

Tip: The velocity of a satellite in orbit is directly dependent on its altitude—the higher the altitude, the lower the orbital speed needed for a stable orbit.