Earth speeds along at \(29.8 \mathrm{km} / \mathrm{s}\) in its orbit. Neptune's nearly circular orbit has a radius of \(4.5 \times 10^{9} \mathrm{km},\) and the planet takes 164.8 years to make one trip around the Sun. Calculate the speed at which Neptune plods along in its orbit.

To determine the speed at which a planet moves in its orbit, first calculate the circumference of the orbit itself. An orbit in the shape of a circle can have its circumference found using the formula: \(C = 2\textbackslash pi r\) Here, \(C\) stands for the circumference, and \(r\) is the radius of the orbit. In the case of Neptune, the radius \(r\) is given as 4.5 × 10^9 km. By plugging in the values, we get: \[C = 2 \textbackslash pi (4.5 \textbackslash times 10^9) \text{km} \approx 28.27 \textbackslash times 10^9 \text{km}\] This calculation provides the total distance Neptune travels in one full orbit.

Planetary motion is how planets orbit the Sun. This motion is not random; it's governed by the laws of physics, particularly Newton's laws of motion and universal gravitation. Planets move in elliptical or nearly circular paths due to the gravitational pull of the Sun. Neptune’s orbit, being nearly circular, can be simplified to calculations involving circular motion. By understanding its circular path, we can better calculate its orbital speed. The constant gravitational pull acting between Neptune and the Sun ensures that Neptune continues to move in its orbit, maintaining a consistent path over a long period.

The orbital radius refers to the distance between the center of the planet's orbit and the center of the Sun. For a nearly circular orbit like the one Neptune has, the radius remains almost constant. In this exercise, Neptune's orbital radius is stated as \(4.5 \times 10^9 \text{km}\). This radius is an essential part of our calculations because it helps determine the total circumference of Neptune’s orbit. Remember, when dealing with planetary orbits, especially in homework problems, knowing the radius allows you to use it in other essential formulas like those for circumference and orbital speed.

To find the speed of a planet in its orbit, we use the formula for speed: \(v = \frac{C}{T}\) where \(v\) stands for speed, \(C\) is the circumference of the orbit, and \(T\) is the orbital period (the time it takes for one full orbit). We have already calculated the circumference \(C = 28.27 \times 10^9 \text{km}\). Neptune's orbital period \(T\) is given as 164.8 years. To use this in our formula, convert years to seconds for consistency in units: \[T = 164.8 \text{years} \times 365.25\text{days/year} \times 24\text{hours/day} \times 3600\text{seconds/hour} \approx 5.195 \times 10^9 \text{seconds}\] Now, calculate the speed \[v = \frac{28.27 \times 10^9 \text{km}}{5.195 \times 10^9 \text{seconds}} \approx 5.44 \text{km/s}\] Hence, Neptune travels at approximately 5.44 km/s in its orbit around the Sun.