Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 km and a mass of 1.0×10¹³ kg hits the earth (6.0×10²⁴ kg) with an impact speed of 4.0×10⁴ m/s. What percentage is this of the earth's speed around the sun? The earth orbits the sun at a distance of 1.5×10¹¹ m .

검증된 단계별 안내

Step 1: Calculate the momentum of the asteroid using the formula for linear momentum: \( p = mv \), where \( m \) is the mass of the asteroid (\( 1.0 \times 10^{13} \; \text{kg} \)) and \( v \) is its impact speed (\( 4.0 \times 10^4 \; \text{m/s} \)).

Step 2: Determine the orbital speed of the Earth around the Sun using the formula for circular orbital velocity: \( v = \sqrt{\frac{GM}{r}} \), where \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \; \text{N·m}^2/\text{kg}^2 \)), \( M \) is the mass of the Sun (\( 1.989 \times 10^{30} \; \text{kg} \)), and \( r \) is the orbital radius of the Earth (\( 1.5 \times 10^{11} \; \text{m} \)).

Step 3: Once the Earth's orbital speed is calculated, compare the asteroid's impact speed to the Earth's orbital speed by finding the ratio: \( \text{Percentage} = \left( \frac{v_{\text{asteroid}}}{v_{\text{Earth}}} \right) \times 100 \).

Step 4: Substitute the known values into the ratio formula: \( v_{\text{asteroid}} = 4.0 \times 10^4 \; \text{m/s} \) and \( v_{\text{Earth}} \) (calculated in Step 2).

Step 5: Simplify the expression to find the percentage of the asteroid's speed relative to the Earth's orbital speed. This will give the final answer.

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주요 개념

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Impact Velocity

Impact velocity refers to the speed at which an object strikes another object. In this context, the asteroid's impact speed of 4.0×10⁴ m/s is crucial for understanding the energy released during the collision. This energy can be calculated using kinetic energy formulas, which are essential for assessing the potential effects of such an impact on Earth.

Orbital Speed of Earth

The orbital speed of Earth around the sun is approximately 29,800 m/s. This speed is determined by the gravitational forces acting between the Earth and the sun, and it is essential for understanding how the asteroid's impact speed compares to the Earth's motion. Calculating the percentage of the asteroid's speed relative to Earth's orbital speed provides insight into the relative scale of the impact.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. In the case of the asteroid, its significant mass and high impact velocity would result in a tremendous amount of kinetic energy, which is critical for evaluating the potential consequences of the impact, such as environmental changes and mass extinctions.