The more powerful the gravitational force of a planet, the greater its escape speed, \(v,\) and the greater the gravitational acceleration, \(g\), at its surface. However, in Table 12.1 , the value for \(v\) is much greater for Uranus than for Earth - but \(g\) is smaller on Uranus than on Earth! How can this be?

The formula for gravitational acceleration (g) at the surface of a planet is given by: g = G * (M / R^2) where G is the gravitational constant (6.674 × 10^-11 N m²/kg²), M is the mass of the planet, and R is the radius of the planet.

The formula for escape speed (v) is given by: v = sqrt(2 * G * M / R) where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

Notice that both formulas have a ratio of mass to radius, which is important for understanding the given observation. If a planet has a high mass and a low radius, its gravitational acceleration is high. If a planet has a high mass but a large radius, its escape speed is relatively high.

We know from the exercise that the escape speed (v) for Uranus is greater than for Earth, and the gravitational acceleration (g) is smaller on Uranus than on Earth. To understand this observation, let's compare the mass and radius of Earth and Uranus. Mass of Earth (M_E) = 5.97 × 10^24 kg Radius of Earth (R_E) = 6,371,000 m Mass of Uranus (M_U) = 86.8 × 10^24 kg Radius of Uranus (R_U) = 25,362,000 m

Let's analyze the mass-to-radius ratios for both planets: M_E / R_E^2 = (5.97 × 10^24 kg) / (6,371,000 m)^2 = 1473.1 kg/m^2 M_U / R_U^2 = (86.8 × 10^24 kg) / (25,362,000 m)^2 = 1075.7 kg/m^2 Since the mass-to-radius ratio is higher for Earth than for Uranus (1473.1 > 1075.7), we can conclude that the gravitational acceleration (g) is greater on Earth than on Uranus.

Now let's analyze the escape speed factors: sqrt(M_E / R_E) = sqrt((5.97 × 10^24 kg) / (6,371,000 m)) = 1.08 × 10^13 kg/m^3 sqrt(M_U / R_U) = sqrt((86.8 × 10^24 kg) / (25,362,000 m)) = 1.62 × 10^13 kg/m^3 Since the square root of the mass-to-radius ratio is higher for Uranus than for Earth (1.62 × 10^13 > 1.08 × 10^13), we can conclude that the escape speed (v) is greater for Uranus than for Earth. In summary, the escape speed of Uranus is greater than that of Earth because the mass of Uranus is much greater, while its radius is also significantly larger - leading to a higher value in the escape speed formula. At the same time, the gravitational acceleration on Uranus is smaller than on Earth because the mass-to-radius ratio is lower for Uranus, which results in a lower value in the gravitational acceleration formula.