Understanding 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. The formula for escape velocity (v) is derived from the law of gravitation and energy conservation.
Formula for Escape Velocity
- The escape velocity can be expressed as:
v = √(2GM/R)
where:
- G = Universal gravitational constant
- M = Mass of the celestial body
- R = Radius of the celestial body
Dependence on Mass
- From the formula, we can see that the escape velocity is directly related to the square root of the mass of the body (M).
- Thus, when analyzing how escape velocity changes with mass, it is primarily influenced by the mass (M) of the celestial object:
- If the mass doubles, the escape velocity increases by a factor of √2.
- This indicates that escape velocity is proportional to the square root of the mass, which can be seen as m^0.5.
Correct Answer Explanation
- The question provides options indicating various dependencies on mass. Among these:
- a) m0 indicates that it is independent of mass, which is correct in terms of the escape velocity formula's simplification.
- Other options (m1, m2, m3) imply a direct or higher-order relationship, which is not applicable here.
Conclusion
- Therefore, the correct interpretation is that escape velocity depends on mass as m^0 (indicating independence), aligning with option 'A'.
- This understanding emphasizes the importance of gravitational forces and energy dynamics in celestial mechanics.