The mass of the sun is 2.0 x 10³⁰ kg. A 5.0 x 10¹⁴ kg comet is 75 million kilometers from the sun. What is the magnitude of the comet's acceleration toward the sun?

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Step 1: Identify the formula for gravitational force, which is given by \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \), \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between them.

Step 2: Use Newton's second law \( F = m \cdot a \) to relate the gravitational force to the comet's acceleration. Rearrange the formula to solve for acceleration: \( a = \frac{F}{m_2} \).

Step 3: Substitute the expression for gravitational force into the acceleration formula: \( a = \frac{G \cdot m_1}{r^2} \). Here, \( m_1 \) is the mass of the sun, \( m_2 \) cancels out, and \( r \) is the distance between the sun and the comet.

Step 4: Convert the distance \( r \) from kilometers to meters. Since \( 1 \, \text{km} = 1000 \, \text{m} \), \( r = 75 \, \text{million kilometers} = 75 \times 10^6 \, \text{km} \times 1000 \, \text{m/km} = 75 \times 10^9 \, \text{m} \).

Step 5: Substitute the values \( G = 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \), \( m_1 = 2.0 \times 10^{30} \, \text{kg} \), and \( r = 75 \times 10^9 \, \text{m} \) into the formula \( a = \frac{G \cdot m_1}{r^2} \). Simplify the expression to find the comet's acceleration.

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주요 개념

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

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is fundamental in calculating gravitational forces, which is essential for determining the acceleration of objects in a gravitational field.

Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the gravitational force exerted by a massive body, such as the sun. It can be calculated using the formula a = F/m, where F is the gravitational force and m is the mass of the object. This concept is crucial for understanding how objects like comets move in space under the influence of gravity.

Distance in Gravitational Calculations

In gravitational calculations, the distance between the centers of two masses is critical, as it affects the gravitational force and, consequently, the acceleration. The distance must be measured in consistent units, typically meters, to ensure accurate calculations. In this question, the distance of 75 million kilometers must be converted to meters to apply the gravitational formulas correctly.