Question

On December 25, 2004, during a NASA mission to Saturn, the spacecraft Cassini released a probe named Huygens, which landed on the Saturnian moon Titan on January 14,2005 . Huygens was released from the main spacecraft at a gentle relative speed of \(31 \mathrm{~cm} / \mathrm{s}\). As Huygens moved away, it rotated at a rate of seven revolutions per minute. (a) How many revolutions had Huygens completed when it was \(150 \mathrm{~m}\) from Cassini? (b) How far did Huygens move away from Cassini during each revolution? Give your answer in meters.

Short answer

(a) 56.45 revolutions. (b) 2.66 meters per revolution.

Step-by-step solution

Calculate Time to Reach 150 m

We know the speed at which Huygens moved away from Cassini is 31 cm/s. First, convert this speed to meters per second: \(31 \text{ cm/s} = 0.31 \text{ m/s}\). We need to find the time taken to reach a distance of 150 meters. Use the formula \( \text{Distance} = \text{Speed} \times \text{Time} \) to solve for time (\( t \)): \[ t = \frac{150 \text{ m}}{0.31 \text{ m/s}} \approx 483.87 \text{ s} \].

Calculate Revolutions in Given Time

Now, we need to find out how many revolutions Huygens completed in 483.87 seconds. Huygens was rotating at a rate of 7 revolutions per minute, which converts to \( \frac{7}{60} \text{ revolutions per second} \). Thus, the total number of revolutions completed is: \[ \text{Revolutions} = \frac{7}{60} \times 483.87 \approx 56.45 \].

Calculate Distance per Revolution

Now we want to find out how far Huygens moved for each revolution. We can use the formula \( \text{Distance per revolution} = \frac{\text{Total distance}}{\text{Total revolutions}} \).\[ \text{Distance per revolution} = \frac{150 \text{ m}}{56.45} \approx 2.66 \text{ meters} \].

Key concepts

Space Missions

Space missions are carefully planned scientific journeys into outer space to explore planets, moons, and other celestial bodies. These missions often involve sending spacecraft equipped with scientific instruments to gather data which can be analyzed to understand more about the universe.

• Spacecraft like the Cassini-Huygens mission to Saturn play a crucial role in gathering information about distant planets.
• Space missions require precise engineering to ensure that spacecraft can survive the harsh environment of space and reach their intended destinations.
• Data collected from these missions can offer insights into the formation and evolution of planets and moons, influencing our understanding of the solar system.

Understanding the specifics of a space mission helps us appreciate the complexity and coordination required, from managing equipment to navigating the vastness of space effectively.

Calculating Revolutions

Calculating revolutions involves determining how many complete rotations an object makes around a point in a given period. This is particularly useful in contexts such as mechanical systems and space missions.

• In the example of the Huygens probe, we calculated how many revolutions it completed after being released from Cassini.
• We converted its rotational speed from revolutions per minute to revolutions per second, as time measurements in physics are often in seconds.
• By multiplying the rotational speed by the total time it was in motion, we found the total number of revolutions.

Being familiar with this calculation helps in understanding rotational dynamics, a foundational concept in physics that appears in many real-world applications.

Relative Speed

Relative speed refers to the velocity of an object as observed from another moving object. It’s an essential concept in physics, especially when analyzing the dynamics of multiple bodies in motion.

• In the Cassini-Huygens mission, the probe was released with a relative speed of 31 cm/s with respect to the main spacecraft.
• Understanding relative speed involves recognizing how the speed of one object is perceived from another object in motion.
• To convert relative speed into different units, like from centimeters per second to meters per second, is crucial for accurate calculations in various problems.

Mastering this concept helps in solving problems involving multiple moving objects, ensuring accurate predictions and analyses of their trajectories and interactions.