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Newton's cannonball is a thought experiment proposed by Sir Isaac Newton. Imagine a cannon placed on a high mountain, powerful enough to shoot a cannonball horizontally at extremely high speeds. If the speed is just right, the cannonball will fall towards Earth due to gravity, but the Earth's surface curves away beneath it at the same rate. This balance of gravitational pull and forward velocity creates a stable orbit.
Newton's idea helped us understand how artificial satellites and other objects like the Moon stay in orbit. By achieving the correct speed (around 7.9 km/s for low Earth orbit), an object can continuously fall towards Earth but never actually hit the ground because the Earth's surface is also curving away under it.

The orbital period is the time it takes for an object to complete one orbit around another object. In this problem, we need to find how long Newton's cannonball takes to orbit Earth once.

The first step is identifying the distance it travels, which is the circumference of its orbit. Given Earth's radius \(r\) of around 6371 km, the circumference (C) can be calculated using:
\[ \text{Circumference} = 2 \pi r \]
Substituting the given radius:
\[ \text{Circumference} = 2 \pi \times 6371 \approx 40030 \text{ km} \]
Next, to find the orbital period (T), we use the formula:
\[ T = \frac{\text{distance}}{\text{speed}} \]
where the distance is 40030 km and the speed is 7.9 km/s. So:
\[ T = \frac{40030 \text{ km}}{7.9 \text{ km/s}} \approx 5066 \text{ seconds} \]
Finally, converting this time into minutes for convenience, we divide by 60:
\[ T \approx \frac{5066 \text{ seconds}}{60 \text{ s/min}} \approx 84.4 \text{ minutes} \]
Therefore, the time for Newton's cannonball to complete one orbit around Earth is approximately 84.4 minutes.

The Earth's circumference is the total distance around the planet at the equator. This is crucial to orbital mechanics calculations as it represents the path length for an object in low Earth orbit. For spherical objects like Earth, the circumference can be found using the formula:
\[ \text{Circumference} = 2 \pi r \]
where \(r\) is the radius of Earth, approximately 6371 km. Plugging this in:
\[ \text{Circumference} = 2 \pi \times 6371 \approx 40030 \text{ km} \]
This value is not just an abstract number. It helps in determining how long an orbit takes and is essential in engineering and space missions. Knowing Earth's circumference allows us to calculate the needed speed and time for objects like satellites to maintain a stable orbit.
Understanding Earth's circumference also plays a pivotal role in navigation, geography, and geodesy, highlighting its importance across multiple scientific fields.