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 revolutions; (b) 2.66 meters per revolution.

Step-by-step solution

Convert Speed to Meters per Second

First, convert the relative speed of Huygens from centimeters per second to meters per second. We know: \(31 \text{ cm/s} = 0.31 \text{ m/s}\).

Determine Time to Reach 150 Meters

Using the speed from the previous step, calculate the time it takes for Huygens to reach a distance of 150 meters using the formula \( t = \frac{d}{v}\), where \(d\) is the distance and \(v\) is the velocity. Thus, \( t = \frac{150}{0.31} \approx 483.87 \text{ s} \).

Convert Time to Minutes

Convert the time from seconds to minutes to match the revolution rate given as revolutions per minute. \( t = \frac{483.87}{60} \approx 8.06 \text{ minutes}\).

Calculate Total Revolutions

Determine how many revolutions Huygens completed in this time period. Given the rotation rate of 7 revolutions per minute, \( ext{total revolutions} = 7 \times 8.06 \approx 56.42 \).

Calculate Distance Per Revolution

Using the total revolutions and the initial distance, calculate how far Huygens traveled per revolution. We calculate as follows: \( ext{distance per revolution} = \frac{150}{56.42} \approx 2.66 \text{ m}\).

Key concepts

Circular Motion

Circular motion involves the movement of an object along the circumference of a circle or circular path. In the context of space probes like Huygens, circular motion can be observed in the rotation behavior of the probe as it travels.
When Huygens was released, it rotated at a rate of 7 revolutions per minute. This means it spun around its axis while moving away from the Cassini spacecraft.

• Revolutions Per Minute (RPM): Measures how many full circles the probe completes in one minute.
• Relationship with Time: As time progresses, the number of circles or revolutions increases proportional to the rotation rate.
• Practical Applications: Helps ensure stability and gather data during travel.

By observing circular motion, scientists are able to maintain control over probes, ensuring they complete their missions successfully.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. In the case of Huygens and Cassini, the relative velocity is the speed at which Huygens moves away from Cassini.
Here, the relative velocity was given as 0.31 m/s. This means that from Cassini's point of view, Huygens was drifting slowly away.

• Importance in Space: Ensures precise navigation between spacecraft and probes.
• Calculations: Using formulas like \( t = \frac{d}{v} \) can determine how long it takes for a probe to reach a certain distance.
• Comparative Analysis: Identifying speed differences between objects in motion.

Understanding relative velocity allows for accurate planning and execution of spacecraft maneuvers.

Space Exploration

Space exploration is the field of science that seeks to understand more about outer space, and it often involves sending spacecraft like Cassini and Huygens to planets and moons.
The mission to Saturn, with the release of the Huygens probe to Titan, is an example of an interplanetary mission aimed at discovering more about our solar system.

• Satellite Missions: Like Cassini-Huygens, they provide valuable data about celestial bodies.
• Scientific Goals: Collect and analyze data about atmospheres, surfaces, and other characteristics of planets or moons.
• Technological Development: Continuously pushes the boundaries for more advanced tools and ships.

Space exploration not only enhances scientific knowledge but also inspires future generations of scientists and engineers.

Saturn

Saturn is the sixth planet from the Sun and is known for its prominent ring system composed of ice and rock particles. During the Cassini mission, Saturn was the target of intense study and observation.
Key reasons why Saturn is a focal point for exploration include:

• Complex Ring System: Offers insight into the physical processes in outer solar systems.
• Many Moons: Titan, its largest moon, was the landing site for the Huygens probe.
• Gas Giant Composition: Consists mainly of hydrogen and helium, offering a window into the formation of the solar system.

Studying Saturn provides vital information about planetary formation and the conditions on its various moons, which might be potential sites for future exploration.