1. The problem is to find the orbital speed of a planet or object orbiting the sun. 2. The formula for orbital speed $v$ in a circular orbit is given by: $$v = \sqrt{\frac{GM}{r}}$$ where: - $G$ is the gravitational constant, - $M$ is the mass of the sun, - $r$ is the radius of the orbit (distance from the sun). 3. Important rules: - The orbit is assumed circular for this formula. - $G$ and $M$ are constants for the solar system. - $r$ must be in meters to keep units consistent. 4. To solve, plug in the values for $G$, $M$, and $r$: $$G = 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2}$$ $$M = 1.989 \times 10^{30} \text{ kg}$$ 5. For example, Earth's average orbital radius is about $r = 1.496 \times 10^{11} \text{ m}$. 6. Calculate the orbital speed: $$v = \sqrt{\frac{6.674 \times 10^{-11} \times 1.989 \times 10^{30}}{1.496 \times 10^{11}}}$$ 7. Simplify inside the square root: $$v = \sqrt{\frac{1.327 \times 10^{20}}{1.496 \times 10^{11}}}$$ 8. Divide the terms: $$v = \sqrt{8.87 \times 10^{8}}$$ 9. Take the square root: $$v \approx 2.98 \times 10^{4} \text{ m/s}$$ 10. So, the orbital speed of Earth around the sun is approximately $29,800$ meters per second. This method applies to any planet or object by substituting its orbital radius $r$.