Question

Imagine that two comets, Comet \(\mathrm{A}\) and Comet \(\mathrm{B},\) are orbiting the Sun. The orbital period for Comet \(\mathrm{A}\) is 50 years, and for Comet \(\mathrm{B},\) it is 60 years. Comet A initially passes at a distance of 600,000 kilometers from Earth, while Comet B initially passes 450,000 kilometers away. At each orbital period, the minimum distances between Earth and the comets decrease by 40,000 kilometers for Comet \(\mathrm{A}\) and 25,000 kilometers for Comet \(\mathrm{B}\). Which comet will impact Earth first?

Short answer

Comet A will impact Earth first after 750 years.

Step-by-step solution

Calculate the Minimum Distance After Each Orbit for Comet A

For Comet A, the initial distance is 600,000 kilometers, and it decreases by 40,000 kilometers each orbit. The distance after one orbit is \(600,000 - 40,000 = 560,000\) km. After two orbits, it's \(560,000 - 40,000 = 520,000\) km, and so on. We continue subtracting 40,000 km for each subsequent orbit until the distance becomes zero or negative.

Determine When Comet A Impacts Earth

To find when Comet A impacts Earth, set up the equation for the distance: \(600,000 - 40,000n = 0\), where \(n\) is the number of orbits. Solve for \(n\):\[600,000 = 40,000n = \frac{600,000}{40,000} = 15\\]This means Comet A will impact Earth after 15 orbits. Since each orbit takes 50 years, impact occurs after \(15 \times 50 = 750\) years.

Calculate the Minimum Distance After Each Orbit for Comet B

For Comet B, the initial distance is 450,000 kilometers, and it decreases by 25,000 kilometers each orbit. The distance after one orbit is \(450,000 - 25,000 = 425,000\) km. Continue subtracting 25,000 km for each subsequent orbit until the distance becomes zero or negative.

Determine When Comet B Impacts Earth

To find when Comet B impacts Earth, set up the equation for the distance: \(450,000 - 25,000m = 0\), where \(m\) is the number of orbits. Solve for \(m\):\[450,000 = 25,000m \m = \frac{450,000}{25,000} = 18\\]This means Comet B will impact Earth after 18 orbits. Since each orbit takes 60 years, impact occurs after \(18 \times 60 = 1080\) years.

Compare Impact Times of Both Comets

Comet A impacts Earth after 750 years, while Comet B impacts after 1080 years. Therefore, Comet A impacts Earth first.

Key concepts

Understanding Orbital Periods

The orbital period of a comet refers to the time it takes for the comet to complete one full orbit around the Sun. This period depends on the comet's path and gravitational influences. In our example, Comet A has an orbital period of 50 years, meaning it circles the Sun once every 50 years. Comet B, on the other hand, completes its circle every 60 years. Orbital periods help us predict the positions of comets in the solar system at any given time. For ease of understanding, think of a race track where Comet A completes a lap faster than Comet B does. This difference in time allows us to estimate when each comet will be closest to Earth again, helping us predict future movements. The importance of knowing the orbital period lies in its ability to estimate when a comet might pose a collision threat to Earth. Scientists can use this data to prepare strategies to minimize risks. In our case, since Comet A has a shorter orbital period, it goes around the Sun more frequently than Comet B.

Calculating Minimum Distance

Minimum distance refers to the closest point a comet comes to Earth during its orbit. Understanding this helps in assessing potential collision risks. Initially, Comet A comes within 600,000 kilometers of Earth, while Comet B is slightly closer at 450,000 kilometers. For each orbit, the minimum distance declines. For Comet A, the distance decreases by 40,000 kilometers with every orbit. Mathematically, this is shown as: after one orbit at 560,000 km, after two orbits at 520,000 km, and so forth. This pattern continues until the comet's path would lead it to contact Earth. Similarly, Comet B’s minimum distance shrinks by 25,000 kilometers per orbit. Interestingly, even though Comet B starts closer, the slower reduction in distance means it takes longer before it reaches a critical point. Calculating these minimal distances at each orbital cycle is crucial for predicting intersection points with Earth's orbit.

Predicting Earth Impact from Comets

Earth impact prediction involves determining when a comet might collide with our planet. In our scenario, we calculated when each comet would cross paths with Earth based on their decreasing minimum distances. For Comet A, setting up and solving the equation for zero distance identifies the impact time. We find that after 15 orbits, the comet will hit Earth. Given its 50-year orbit, this equates to a 750-year timeline for impact. With Comet B, a similar process shows that the comet doesn't impact Earth until 18 orbits are completed, totaling 1080 years due to its 60-year period. This comparative analysis allows us to determine which comet impacts first, showing the importance of understanding both orbital patterns and distance shrinkage in impact prediction. Thus, by understanding these interactions, scientists can issue warnings and develop measures to prevent or mitigate possible impacts, ensuring Earth's safety.