Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

Verified step by step guidance

1. Identify the formula for escape speed, which is given by \( v_{escape} = \sqrt{\frac{2GM}{R}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.
2. Retrieve the values of \( G \), the mass \( M \) of Mars and Jupiter, and their radii \( R \) from Appendix F or a reliable source.
3. Substitute the values for Mars into the escape speed formula to calculate the escape speed from the surface of Mars.
4. Substitute the values for Jupiter into the escape speed formula to calculate the escape speed from the surface of Jupiter.
5. Explain why the escape speed is independent of the spacecraft's mass: The formula for escape speed does not include the mass of the spacecraft, indicating that the escape speed depends only on the properties of the planet (its mass and radius) and not on the object attempting to escape its gravitational pull.

Key Concepts

Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from a celestial body's gravitational pull without any additional propulsion. It depends on the mass of the body and the distance from its center, calculated using the formula v = √(2GM/r), where G is the gravitational constant, M is the mass of the celestial body, and r is the radius from the center of the body.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This force is what determines the escape velocity, as a stronger gravitational pull requires a higher speed to escape.

Independence from Mass

The escape speed is independent of the mass of the spacecraft because it is derived from the gravitational potential energy and kinetic energy principles. When calculating escape velocity, the mass of the object cancels out, meaning that all objects, regardless of their mass, require the same speed to escape a gravitational field, assuming no air resistance or other forces act on them.