Final answer:
The larger a planet's mass with the same radius, the higher the escape velocity due to the increase in surface gravity. As the mass increases, the gravitational force becomes stronger, requiring a greater speed to escape the gravitational pull.
Step-by-step explanation:
For the same radius, the larger a planet's mass, the higher the escape velocity. Escape velocity is the minimum speed needed for an object to 'break free' from the gravitational attraction of a massive body, without further propulsion. This velocity depends on the mass of the planet and the distance from its center of gravity. In the context of planets, as the mass increases while the radius remains constant, the gravity at the surface becomes stronger, thereby requiring a higher escape velocity.
Looking at the hypothetical planets mentioned, we can see that as the mass increases beyond a certain point, there's an intensification in gravitational compression, which can make the planet smaller. This demonstrates that gravity not only affects escape velocity but also the physical characteristics of astronomical bodies. An example of this is how a red giant has a lower escape velocity at its surface due to an increased radius, whereas the Sun, if compressed, would have a stronger surface gravity, thus a higher escape velocity.
Furthermore, the escape velocity from a celestial body is exactly √2 times greater than the orbital velocity at any given distance from the center of that body, showing a proportional relationship between these two velocities.