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?

검증된 단계별 안내

To calculate the escape speed, use the formula: \( v_{e} = \sqrt{\frac{2GM}{R}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

For part (a), substitute the mass and radius of Mars into the escape speed formula. Use the values: \( M_{Mars} = 6.42 \times 10^{23} \text{ kg} \) and \( R_{Mars} = 3.39 \times 10^{6} \text{ m} \).

For part (b), substitute the mass and radius of Jupiter into the escape speed formula. Use the values: \( M_{Jupiter} = 1.90 \times 10^{27} \text{ kg} \) and \( R_{Jupiter} = 6.99 \times 10^{7} \text{ m} \).

Calculate the escape speed for each planet by plugging the respective values into the formula and solving for \( v_{e} \).

For part (c), understand that the escape speed is independent of the spacecraft's mass because the formula \( v_{e} = \sqrt{\frac{2GM}{R}} \) does not include the mass of the spacecraft. The escape speed depends only on the gravitational parameters of the planet.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Escape Speed

Escape speed is the minimum speed needed for an object to break free from the gravitational pull of a celestial body without further propulsion. It depends on the mass and radius of the body being escaped from, calculated using the formula v = sqrt(2GM/R), where G is the gravitational constant, M is the mass of the celestial body, and R is its radius.

Gravitational Constant

The gravitational constant (G) is a fundamental constant in physics that quantifies the strength of the gravitational force between two masses. Its value is approximately 6.674 × 10^-11 N(m/kg)^2. It is crucial in calculating gravitational forces and escape speeds, as it relates the mass and distance between objects to the force exerted.

Independence from Mass

The escape speed is independent of the mass of the spacecraft because the gravitational force and the required kinetic energy both scale with mass. As the mass increases, the gravitational pull increases proportionally, but so does the kinetic energy needed to escape, resulting in mass canceling out in the escape speed formula, leaving it dependent only on the celestial body's properties.