R48 · Low-orbit satellite: speed and period (orbital/Kepler estimate)

This problem requires calculating the orbital radius first, then using that value to find the speed and finally the period.

### Given Values:
*   Earth's Radius ($R_E$): $6371 \text{ km}$
*   Altitude ($h$): $400 \text{ km}$
*   Standard Gravitational Parameter ($\mu$): $398600 \text{ km}^3/\text{s}^2$

***

### Step 1: Calculate the Orbital Radius ($r$)

The orbital radius is the sum of Earth's radius and the altitude.
$$r = R_E + h$$
$$r = 6371 \text{ km} + 400 \text{ km}$$
$$\mathbf{r = 6771 \text{ km}}$$

***

### (a) Estimate the Orbital Speed ($v$)

We use the formula for orbital speed in

To solve this problem, we need to follow these steps:

### (a) Orbital Speed
The formula for orbital speed \( v \) is given by:
\[ v = \sqrt{\frac{\mu}{r}} \]
where:
- \(\mu\) is the standard gravitational parameter (\(\approx 398600 \, \text{km}^3/\text{s}^2\))
- \( r \) is the distance from the Earth's center to the satellite.

First, we calculate \( r \):
\[ r = R_{\text{Earth}} + h \]
where:
- \( R_{\text{Earth}} \) is the radius of the Earth (\(\approx 6371 \, \text{km}\))
- \( h \) is the altitude of the satellite (400 km)

So,
\[ r = 6371 \, \text{km} + 400 \, \text{km} = 6771 \, \text{km} \]

Now, we can find the orbital speed:
\[ v = \sqrt{\frac{398600 \, \text{km}^3/\text{s}^2}{6771 \, \text{km}}} \]
\[ v = \sqrt{\frac{398600}{6771}} \]
\[ v \approx \sqrt{58.87} \]
\[ v \approx 7.67 \, \text{km/s} \]

### (b) Orbital Period
The formula for the orbital period \( T \) is given by:
\[ T = \frac{2\pi r}{v} \]
where:
- \( r \) is the distance from the Earth's center to the satellite (\( 6771 \, \text{km} \))
- \( v \) is the orbital speed (\(\approx 7.67 \, \text{km/s}\))

First, we convert \(\pi\) to approximately 3.14159.

Now, we can calculate the period:
\[ T = \frac{2 \times 3.14159 \times 6771 \, \text{km}}{7.67 \, \text{km/s}} \]
\[ T \approx \frac{42498.6}{7.67} \]
\[ T \approx 5540.8 \, \text{s} \]

To convert seconds to minutes:
\[ T \approx \frac{5540.8 \, \text{s}}{60} \approx 92.35 \, \text{minutes} \]

### Final Numbers
- **Orbital Speed:** \( \approx 7.67 \, \text{km/s} \)
- **Orbital Period:** \( \approx 92.35 \, \text{minutes} \)

This problem requires calculating the orbital speed using the energy equation and then using that speed to determine the period.

### Given Values
*   Altitude ($h$): $400 \text{ km}$
*   Earth Radius ($R_E$): $6371 \text{ km}$
*   Standard Gravitational Parameter ($\mu$): $398600 \text{ km}^3/\text{s}^2$

***

### Step 1: Calculate the Orbital Radius ($r$)
First, we must find the total distance from the center of Earth to the satellite.
$$r = R_E + h$$
$$r = 6371 \text{ km} + 400 \text{ km}$$
$$\mathbf{r = 6771 \text{ km}}$$

***

### (a) Estimate the Orbital Speed ($v$)

We use the formula:
$$v = \sqrt{\frac{\mu}{r}}$$

**Show Work:**
$$v = \sqrt{\frac{398600 \text{ km}^3/\text{s}^2}{6771 \text{ km}}}$$
$$v = \sqrt{58.8643 \text{ km}^2/\text{s}^2}$$
$$\mathbf{v \approx 7.672 \text{ km/s}}$$

***

### (b) Estimate the Orbital Period ($T$)

We use the formula:
$$T = \frac{2\pi r}{v}$$

**Show Work:**
$$T = \frac{2\pi (6771 \text{ km})}{7.672 \text{ km/s}}$$
$$T = \frac{42549.8 \text{ km}}{7.672 \text{ km/s}}$$
$$\mathbf{T \approx 5546.0 \text{ seconds}}$$

**Convert to Minutes:**
Since the question asks for the period in minutes, we divide by $60$:
$$T_{\text{minutes}} = \frac{5546.0 \text{ s}}{60 \text{ s/min}}$$
$$\mathbf{T_{\text{minutes}} \approx 92.43 \text{ minutes}}$$

***

### Final Numbers

(a) The orbital speed is: $\mathbf{7.67 \text{ km/s}}$

(b) The orbital period is: $\mathbf{92.43 \text{ minutes}}$

## Step 1: Calculate the radius of the satellite's orbit
First, we need to calculate the total radius of the satellite's orbit by adding the Earth's radius and the altitude of the satellite. The Earth's radius is given as approximately 6371 km, and the satellite's altitude is given as 400 km.

## Step 2: Perform the calculation for the orbital radius
r = Earth's radius + Satellite's altitude
r = 6371 km + 400 km
r ≈ 6771 km

## Step 3: Calculate the orbital speed using v = sqrt(μ/r)
Now, we can use the formula v = sqrt(μ/r) to calculate the orbital speed of the satellite. We are given μ ≈ 398600 km^3/s^2.

## Step 4: Perform the calculation for the orbital speed
v = sqrt(μ/r)
v = sqrt((398600 km^3/s^2)/(6771 km))
v ≈ sqrt(58767.5 km^2/s^2)
v ≈ 242.87 km/s

## Step 5: Calculate the orbital period using T = 2*pi*r/v
Now that we have the orbital speed, we can use the formula T = 2*pi*r/v to calculate the orbital period of the satellite.

## Step 6: Perform the calculation for the orbital period in seconds
T = (2 * π * r) / v
T ≈ (6.28319 * 6771 km) / 242.87 km/s
T ≈ 13779.9 s

## Step 7: Convert the orbital period from seconds to minutes
To convert the orbital period from seconds to minutes, we divide by 60.

## Step 8: Perform the calculation for the orbital period in minutes
T ≈ 13779.9 s / 60 s/min
T ≈ 229.99 min

The final answer is: $\boxed{243}$

To solve for the orbital speed and period, we first need to determine the orbital radius ($r$), which is the distance from the center of the Earth to the satellite.

**1. Calculate the orbital radius ($r$):**
The orbital radius is the sum of the Earth's radius ($R$) and the altitude ($h$).
$$r = R + h$$
$$r = 6371\text{ km} + 400\text{ km} = 6771\text{ km}$$

---

**(a) Calculate the orbital speed ($v$):**
Using the formula $v = \sqrt{\frac{\mu}{r}}$:

$$v = \sqrt{\frac{398600\text{ km}^3/\text{s}^2}{6771\text{ km}}}$$
$$v = \sqrt{58.8687\dots\text{ km}^2/\text{s}^2}$$
$$v \approx 7.67\text{ km/s}$$

**Orbital speed $\approx 7.67\text{ km/s}$**

---

**(b) Calculate the orbital period ($T$):**
Using the formula $T = \frac{2\pi r}{v}$ to find the period in seconds, then converting to minutes:

First, find the circumference of the orbit:
$$\text{Circumference} = 2 \cdot \pi \cdot 6771\text{ km} \approx 42543.45\text{ km}$$

Now, calculate $T$ in seconds:
$$T = \frac{42543.45\text{ km}}{7.6726\text{ km/s}} \approx 5544.85\text{ seconds}$$

Convert seconds to minutes:
$$T_{\text{minutes}} = \frac{5544.85}{60}$$
$$T_{\text{minutes}} \approx 92.41\text{ minutes}$$

**Orbital period $\approx 92.41\text{ minutes}$**

This problem requires calculating the orbital speed and subsequently the orbital period using basic orbital mechanics formulas.

### Given Values:
*   Altitude ($h$): $400 \text{ km}$
*   Earth's Radius ($R_E$): $\approx 6371 \text{ km}$
*   Standard Gravitational Parameter ($\mu$): $398600 \text{ km}^3/\text{s}^2$

---

### Step 1: Calculate the Orbital Radius ($r$)
The orbital radius is the distance from the center of Earth to the satellite.
$$r = R_E + h$$
$$r = 6371 \text{ km} + 400 \text{ km}$$
$$\mathbf{r = 6771 \text{ km}}$$

### (a) Estimate the Orbital Speed ($v$)

We use the formula: $v = \sqrt{\mu/r}$

$$v = \sqrt{\frac{398600 \text{ km}^3/\text{s}^2}{6771 \text{ km}}}$$
$$v = \sqrt{58.8704 \dots \text{ km}^2/\text{s}^2}$$
$$\mathbf{v \approx 7.6726 \text{ km/s}}$$

***

### (b) Estimate the Orbital Period ($T$)

We use the formula: $T = 2\pi r / v$

*Note: Since we are calculating the period in minutes, we must ensure all units cancel correctly.*

$$T = \frac{2\pi (6771 \text{ km})}{7.6726 \text{ km/s}}$$
$$T \approx 5543.2 \text{ seconds}$$

Now convert the period from seconds to minutes:
$$T_{\text{minutes}} = T_{\text{seconds}} / 60$$
$$T_{\text{minutes}} = 5543.2 \text{ s} / 60 \text{ s/min}$$
$$\mathbf{T \approx 92.39 \text{ minutes}}$$

---

### Final Estimates:

**(a) The orbital speed is:** $\mathbf{7.67 \text{ km/s}}$ (or $7.673 \text{ km/s}$)

**(b) The orbital period is:** $\mathbf{92.39 \text{ minutes}}$

Here is how we can estimate the answers to this question:

a) First, let us compute the distance of the satellite from Earth's center, $r$:

$$
\begin{equation*}
    r = 6371 \text{ km} + 400 \text{ km} \\
    = 6771 \text{ km}\,.
\end{equation*}
$$

Now we can calculate the orbital speed $v$ using: 

$$ 
\begin{align*}
    v &= \sqrt{\frac{\mu}{r}}\\ 
    &= \sqrt{\frac{398600 \text{ km}^3/\text{s}^2}{6771 \text{ km}}} \\
    &\approx 7.54 \text{ km/s}\,.
\end{align*} 
$$ 

b) To find the orbital period $T$ in minutes, we'll first calculate it in seconds and then convert: 

$$
\begin{align*} 
    T &= \frac{2\pi r}{v} \\
    &= \frac{2\pi (6771 \text{ km})}{7.54 \text{ km/s}} \\
    &\approx 5380.39 \text{ s}\,.
\end{align*}  
$$ 

Now we convert to minutes: 

$$
\begin{equation*} 
     T = \frac{5380.39 \text{ s}}{60 \text{ s/min}} \\
     \approx 89.7 \text{ min}\,.
\end{equation*}  
$$ 

So, the orbital speed is approximately $7.54$ km/s and the orbital period is around $89.7$ minutes.

<think>
Okay, so I need to figure out the orbital speed and the orbital period of a satellite that's orbiting Earth at an altitude of 400 km. Let me start by recalling what the problem said. They gave Earth's radius as approximately 6371 km and the gravitational parameter μ as 398600 km³/s². The formulas provided are for velocity, v = sqrt(μ/r), and for the period, T = 2πr/v. Alright, let me break this down step by step.

First, I need to find the orbital speed. The formula requires the distance from the center of the Earth, which is the Earth's radius plus the altitude of the satellite. The satellite is at 400 km above the Earth's surface, so the total distance r would be Earth's radius (6371 km) plus 400 km. Let me calculate that:

r = 6371 km + 400 km = 6771 km.

Hmm, okay. So that's the value of r. Now, plugging this into the velocity formula:

v = sqrt(μ / r)

So substituting the given μ value and the r we just found:

v = sqrt(398600 km³/s² / 6771 km)

Wait, let me make sure the units work out. Since μ is in km³/s² and r is in km, dividing them would give (km³/s²) / km = km²/s², so taking the square root gives km/s, which is what we want for velocity. That looks good.

Let me compute that value. Let me first compute the denominator: 6771 km. Let me write the division part first:

398600 divided by 6771. Let me do that division. Hmm, maybe I can approximate or use a calculator step here. Let me see:

398600 / 6771 ≈ ?

Let me see, 6771 times 58 is 6771*50=338,550 and 6771*8=54,168. So 338,550 +54,168 = 392,718. That's 58 times 6771. Hmm, but 392,718 is less than 398,600. The difference is 398,600 - 392,718 = 5,882. So then 5,882 divided by 6771 ≈ ~0.87. So total would be approximately 58.87?

Wait, let me check that again. Wait, 58 times 6771 is 58*6000=342,000; 58*700=40,600; 58*71≈4,118. So adding those together: 342,000 + 40,600 = 382,600 + 4,118 = 386,718? Wait, maybe my previous calculation was off. Hmm, perhaps better to do exact division.

Alternatively, use calculator steps:

Let me compute 398600 divided by 6771. Let me write this as:

Dividend = 398,600

Divisor = 6,771

Let me see how many times 6,771 goes into 398,600.

First, 6771 x 58 = let's see, 6771*50 = 338,550; 6771*8 = 54,168. So 338,550 + 54,168 = 392,718.

Subtract that from 398,600: 398,600 - 392,718 = 5,882.

So now, 5,882 divided by 6,771 is approximately ~0.87. So total is 58.87.

Therefore, 398600 / 6771 ≈ 58.87 (km²/s²). Wait, but actually, the units would be (km³/s² divided by km) gives km²/s², so the square root is sqrt(km²/s²) = km/s. So the velocity would be sqrt(58.87). Let me compute that.

What is sqrt(58.87)? Let me see, 7^2 is 49, 8^2 is 64, so sqrt(58.87) is between 7.6 and 7.7.

Let me compute 7.67^2: 7.6^2 = 57.76. 7.67^2 = (7.6 + 0.07)^2 = 7.6^2 + 2*7.6*0.07 + 0.07^2 = 57.76 + 1.064 + 0.0049 = 58.8289. That's really close to 58.87. So sqrt(58.87) ≈ 7.67 + a little. The difference between 58.87 and 58.8289 is 0.0411. To get a more precise value, maybe approximate:

Let me let x = 7.67, then (x + δ)^2 = x² + 2xδ + δ² = 58.8289 + 2*7.67*δ + δ². We need this to equal 58.87, so 2*7.67*δ ≈ 58.87 - 58.8289 ≈ 0.0411.

So δ ≈ 0.0411 / (2*7.67) ≈ 0.0411 / 15.34 ≈ ~0.0027. So δ ≈ 0.0027. So sqrt(58.87) ≈ 7.67 + 0.0027 ≈ 7.6727. So approximately 7.673 km/s.

Therefore, the orbital speed is approximately 7.67 km/s. Hmm, let me verify if that seems reasonable. I remember that the International Space Station orbits at around 400 km, and its speed is roughly 7.8 km/s, so this is a bit lower. Wait, is that right? Hmm, maybe I made an error in calculation.

Wait, let's double-check the calculation for 398600 divided by 6771.

Wait 6771 * 58.87 = ?

Wait perhaps using calculator steps:

Let me try to compute 398600