https://file.aiquickdraw.com/mathbot/file/17339042721579xk3eznx.jpg

The formula displayed in the image is:

\[
v = \sqrt{\frac{6.674 \times 10^{-11} \, \text{m}^3/(\text{kg}\cdot\text{s}^2) \times 5.972 \times 10^{24} \, \text{kg}}{8000.0 \times 10^3 \, \text{m}}}
\]

This formula is derived from **Newton's Law of Gravitation** and is used to calculate the orbital velocity \(v\) of an object in a circular orbit around a larger mass (like the Earth). Here's how to solve it step-by-step:

---

### Step 1: Plug in the constants
1. Gravitational constant (\(G\)): \(6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}\)
2. Mass of the Earth (\(M\)): \(5.972 \times 10^{24} \, \text{kg}\)
3. Orbital radius (\(r\)): \(8000.0 \times 10^3 \, \text{m} = 8.0 \times 10^6 \, \text{m}\)

---

### Step 2: Compute the numerator
\[
\text{Numerator} = G \cdot M = (6.674 \times 10^{-11}) \cdot (5.972 \times 10^{24})
\]
\[
\text{Numerator} = 3.986 \times 10^{14}
\]

---

### Step 3: Compute the denominator
\[
\text{Denominator} = r = 8.0 \times 10^6
\]

---

### Step 4: Divide numerator by denominator
\[
\frac{\text{Numerator}}{\text{Denominator}} = \frac{3.986 \times 10^{14}}{8.0 \times 10^6}
\]
\[
= 4.9825 \times 10^7
\]

---

### Step 5: Take the square root
\[
v = \sqrt{4.9825 \times 10^7} \approx 7.06 \times 10^3 \, \text{m/s}
\]

---

### Final Answer:
\[
v \approx 7060 \, \text{m/s}
\]

This is the orbital velocity of an object at a radius of 8000 km from the Earth's center.

---

Would you like me to explain any specific steps further?  

---

Here are five questions to expand understanding:
1. What is the relationship between orbital velocity and orbital radius?
2. How does the mass of the orbiting object affect the velocity calculation?
3. Why is the gravitational constant (\(G\)) so small?
4. Can this formula be applied to elliptical orbits? Why or why not?
5. What happens to the orbital velocity if the altitude of the orbit increases?

**Tip**: Remember that the orbital radius includes the Earth's radius plus the altitude of the orbit.

Calculate the orbital velocity of an object at a radius of 8000 km from the Earth's center using Newton's law of gravitation.

Discover how to calculate the orbital velocity of an object 8000 km from Earth's center. Learn step-by-step how to apply Newton's law of gravitation, including calculations for the gravitational constant, Earth's mass, and orbital radius. Perfect for high school physics students.

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