**Understanding Orbital Velocity and Signal Time Ratios for Mercury and the Moon***Problem Overview:*1. **Orbital Velocity of a Probe Around Mercury:** - **Given:** A probe orbits Mercury at 192 km above the surface. - **Objective:** Calculate the orbital velocity required to maintain orbit. - **Data:** - Mass of Mercury: \(3.30 \times 10^{23} \) kg - Radius of Mercury: \(2.44 \times 10^3 \) km2. **Signal Time Ratio from Earth:** - **Objective:** Find the ratio of the time it takes a signal to reach Mercury compared to the Moon. - **Distances:** - Earth to Mercury: \(57.9 \times 10^6\) km - Earth to Moon: 384,400 km3. **Wavelength Shift Calculation:** - **Given:** Signal wavelength at 15 cm. - Determine wavelength shift at the calculated orbital velocity. Assume the probe is directly moving away from Earth.**Part 1 of 4: Orbital Velocity Calculation**- **Formula:** \[ v_c = \sqrt{\frac{GM}{r}} \] - \( v_c \): Orbital velocity - \( G \): Gravitational constant - \( M \): Mass of Mercury - \( r \): Distance from the center of Mercury (radius + altitude)- **Calculation:** \[ r = 2.44 \times 10^3 \text{ km} + 192 \text{ km} = 2.63 \times 10^6 \text{ m} \] \[ v_c = \sqrt{\frac{6.67 \times 10^{-11} \times 3.30 \times 10^{23}}{2.63 \times 10^6}} = 2.89 \text{ km/s} \]**Part 2 of 4: Signal Time Ratio**- **Concept:** Signal travel time is connected to distance over speed of light. \[ t = \frac{d}{v} \]- **Expressions:** - Time to Mercury: \[ t_{\text{Mercury}} = \frac{d_{\text{Merc

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You send a probe to orbit Mercury at 192 km above the surface. What orbital velocity (in km/s) is needed to keep it in orbit? (The mass of Mercury is 3.30 x 1023 kg, and the radius of Mercury is 2.44 x 103 km.) What is the ratio of the time it takes a signal from Earth to reach Mercury (d = 57.9 × 106 km) to the time it would take to reach the Moon (d = 384,400 km)? If your signal is at 15 cm, what is the wavelength shift (in cm) at this orbital velocity? (Assume the probe is at a point in its orbit in which it is moving directly away from the Earth.)

You send a probe to orbit Mercury at 192 km above the surface. What orbital velocity (in km/s) is needed to keep it in orbit? (The mass of Mercury is 3.30 x 1023 kg, and the radius of Mercury is 2.44 x 103 km.) What is the ratio of the time it takes a signal from Earth to reach Mercury (d = 57.9 × 106 km) to the time it would take to reach the Moon (d = 384,400 km)? If your signal is at 15 cm, what is the wavelength shift (in cm) at this orbital velocity? (Assume the probe is at a point in its orbit in which it is moving directly away from the Earth.)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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