Escape velocity is the minimum speed an object must reach to break free from a celestial body’s gravitational influence. For Earth, the escape velocity is approximately 11.2 km/s. This value is determined by the mass and radius of the planet. When considering changes to these parameters, the escape velocity will also change.

If the mass of the Earth doubles while the radius is halved, the implications for escape velocity are significant. A doubling of the Earth’s mass increases the gravitational force exerted on an object at its surface. Concurrently, reducing the radius decreases the distance from the center of the Earth to the surface, which also increases the gravitational force felt by an object.

As a result of these changes, the new escape velocity becomes approximately 22.4 km/s. This is double the original escape velocity. Therefore, a body must now reach this higher speed to escape Earth’s gravitational pull. This scenario highlights the relationship between mass, radius, and escape velocity, illustrating how modifications in these parameters can lead to significant changes in the energy required for an object to break free from a planet’s gravitational field.

The escape velocity of a celestial body is the minimum speed required for an object to break free from its gravitational influence without any further propulsion. For a planet with a mass equal to that of Earth but with a radius that is one-fourth of Earth’s radius, the escape velocity is much higher than that of Earth.

This is due to the relation between mass and radius in gravitational physics. The gravitational pull an object experiences on the surface of the planet depends on both the mass of the planet and the distance from its center. As the radius is reduced to one-fourth, the gravitational force at the surface increases, and it needs a greater speed to escape the gravitational field of the planet.

This means that the escape velocity for this hypothetical planet is about 22.4 km/s, which is double the escape velocity of Earth, which is about 11.2 km/s. This means that an object launched from the surface of this planet must achieve a much higher speed to overcome the stronger gravitational attraction. Understanding this concept is crucial for space missions and exploring the dynamics of celestial bodies in our universe.

When a planet revolves in an elliptical orbit around the sun, a variety of energies and angular momentums are involved. The total energy, E in this context, is thus very important since it stands for the sum of kinetic energy T and gravitational potential energy V of the planet.

In any bound elliptical orbit of a planet, total energy is always negative. This negative value means that the planet is gravitationally bound to the sun, with insufficient energy to leave the influence of the gravitational pull from the sun. The reason why the gravitational potential energy is negative is because of the attraction caused by gravity; the more negative this energy gets as the planet approaches the sun, the greater is its kinetic energy, which peaks at periapsis, the closest point of the orbit to the sun.

On the other hand, at apoapsis, where the planet is farthest from the sun, its kinetic energy is lower, but the gravitational potential energy is still negative. The total energy is consistently negative, and this is what reflects the stability of the orbit. Therefore, it is evident that for a planet in an elliptical orbit, the total energy is always negative, which is a further proof of the concept of gravitational binding in celestial mechanics.