Problem 65 - University Physics with Modern Physics

Question

Trip to Mars. You are working on a project with NASA to launch a rocket to Mars, with the rocket blasting off from earth when earth and Mars are aligned along a straight line from the sun. If Mars is now \(60^{\circ}\) shead of earth in its orbit around the sun, when should you launch the rocket? (Note: All the planets orbit the sun in the same direction, 1 year on Mars is 1.9 earth- years, and assume circular orbits for both planets.)

Step-by-Step Solution

Answer

Launch the rocket in 577 days from now.

Step 1: Determine Orbital Parameters

First, understand that Earth completes its orbit in 1 year and Mars in 1.9 Earth-years due to Kepler's third law. This means that the angular velocity \( \omega \) of Earth around the sun is \( \omega_E = \frac{360^{\circ}}{1} \) and for Mars it's \( \omega_M = \frac{360^{\circ}}{1.9} \).

Step 2: Set Up the Relative Position Equation

The goal is to launch the rocket when Earth and Mars are aligned in a straight line relative to the Sun. Initially, Mars is \(60^{\circ}\) ahead. As Earth travels faster, it needs to 'catch up'. The relative angular position \( \theta(t) \) over time \( t \) in Earth-years can be modeled by \( \theta(t) = (\omega_M - \omega_E) \cdot t + 60^{\circ} \). For them to be aligned, \( \theta(t) = 360^{\circ} \times N \) where \( N \) is an integer.

Step 3: Solve for Time Interval \( t \)

Replace \( \theta(t) \) with \( 360^{\circ} \) (since we are looking for the first time they align): \[ (\omega_M - \omega_E) \cdot t + 60^{\circ} = 360^{\circ} \] Replace \( \omega_M \) and \( \omega_E \):\[ \left(\frac{360^{\circ}}{1.9} - 360^{\circ}\right)t + 60^{\circ} = 360^{\circ} \] \[ \frac{-168^{\circ}}{1.9}t + 60^{\circ} = 360^{\circ} \] Solve for \( t \): \[ -189.47t + 60 = 360 \] \[ -189.47t = 300 \] \[ t = \frac{300}{-189.47} \approx 1.58 \text{ years} \].

Step 4: Calculate Launch Time in Days

Convert the years to days: 1 Earth-year is 365 days, so \( t \) is approximately \( 1.58 \times 365 \approx 577 \) days.

Step 5: Summary and Conclusion

You should launch the rocket approximately 577 days from now to ensure Earth and Mars align for a direct path.

Key Concepts

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space under the influence of gravitational forces. It forms the basis of our understanding of how celestial bodies, like planets and satellites, travel in their paths. In our exercise, this concept helps us determine when and how to launch a rocket to Mars, considering both planets are orbiting the sun in circular paths. Orbital mechanics relies heavily on mathematical principles and laws of physics to calculate trajectories and predict future positions of celestial bodies.

• Gravitational forces: the primary factor driving orbital paths.
• Planetary orbits: assumed circular for simplification, but are generally elliptical.
• Understanding motion: allows prediction of how Earth and Mars move relative to each other.

These principles allow us to plan space missions accurately, ensuring rockets are launched at the right moment when the planets align correctly.

Kepler's Third Law

Kepler's Third Law is crucial in the field of astrophysics and celestial mechanics. It states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Simply put, planets further from the sun take longer to complete their orbit than those closer.
In our problem of reaching Mars, Kepler's Third Law helps us understand why Earth and Mars have different orbital periods – Earth orbits in 1 year, while Mars takes about 1.9 Earth-years. This difference is essential for planning the time to launch a rocket, as it impacts the relative positions of Earth and Mars in their orbits.

• Orbital period: the time it takes for a planet to complete one orbit around the sun.
• Semi-major axis: one of the half-axes of an elliptical orbit, but approximated here as the orbital radius in circular orbits.
• Practical application: used to calculate travel time between planets.

This law provides the foundation for synchronizing movements of different planets, essential when venturing into interplanetary travel.

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point. In our scenario, it describes how quickly Earth and Mars move around the sun. Angular velocity, denoted by \(\omega\), is calculated by dividing the total angle of the orbit (360 degrees) by the orbital period.
For Earth, \(\omega_E = \frac{360^{\circ}}{1} = 360^{\circ}\) per year, and for Mars, \(\omega_M = \frac{360^{\circ}}{1.9}\). This tells us that Earth moves quicker along its orbit compared to Mars.

• Utility: determines how fast celestial bodies rotate in their orbits.
• Comparison: Earth's higher angular velocity means it traverses its orbit faster than Mars.
• Significance: crucial for calculating when Earth 'catches up' with Mars.

Understanding angular velocity helps to align planetary positions for efficient space travel, such as launching a rocket when planets align relative to another point like the sun.

Relative Angular Position

Relative angular position is about where two objects are located in their orbits at any given time. In this exercise, it involves comparing the positions of Earth and Mars as they travel around the sun.
Initially, Mars is given as being \(60^{\circ}\) ahead of Earth. This initial position is vital when calculating the exact launch time. As time progresses, Earth, with a faster angular velocity, changes this relative position, gradually catching up to Mars. The desired alignment occurs when this position angle reaches a full circle (360 degrees), making the planets inline from the sun’s perspective.

• Initial condition: Mars is 60° ahead of Earth.
• Alignment goal: achieve a 360° relative position for successful rocket launch.
• Mathematical modeling: \(\theta(t) = (\omega_M - \omega_E) \cdot t + 60^{\circ}\).

Accurately predicting when this alignment occurs enables precise planning of interplanetary missions, ensuring fuel efficiency and successful arrival at the destination.