Approaching the ISS A Russian Soyuz module, with three...

Approaching the ISS A Russian Soyuz module, with three astronauts and a full load of cargo, has a mass of 7500 kg. The International Space Station (ISS) has a mass of $420,000$ kg. When Soyuz docks at the ISS, the two centers of mass are separated by 9.5 (a) Find the average of the gravitational force of attraction between the two spacecraft when Soyuz is docked and when it is 110 $\mathrm{m}$ from the ISS. (b) If Soyuz approached the ISS from rest a distance of $110 \mathrm{m},$ a thruster would have to counteract the average force of gravitational attraction between it and the ISS. If this thruster has an exhaust velocity of $590 \mathrm{m} / \mathrm{s},$ at what average rate must it burn fuel (in $\mathrm{kg} / \mathrm{s}$ ) to counteract the pull of the ISS? (See Section $9-8$ for a discussion of thrust. $(\mathrm{c})$ If the approach requires $330 \mathrm{s},$ how much fuel will the thruster burn in order to counteract the gravitational attraction?

Approaching the ISS A Russian Soyuz module, with three astronauts and a full load of cargo, has a mass of 7500 kg. The International Space Station (ISS) has a mass of $420,000$ kg. When
Soyuz docks at the ISS, the two centers of mass are separated by 9.5 (a) Find the average of the gravitational force of attraction between the two spacecraft when Soyuz is docked and when it is
110 $\mathrm{m}$ from the ISS. (b) If Soyuz approached the ISS from rest a distance of $110 \mathrm{m},$ a thruster would have to counteract the average force of gravitational attraction between it and the ISS. If this thruster has an exhaust velocity of $590 \mathrm{m} / \mathrm{s},$ at what average rate must it burn fuel (in $\mathrm{kg} / \mathrm{s}$ ) to counteract the pull of the ISS? (See Section $9-8$ for a discussion of thrust. $(\mathrm{c})$ If the approach requires
$330 \mathrm{s},$ how much fuel will the thruster burn in order to counteract
the gravitational attraction?

Key Concepts

Newton's Law of Universal Gravitation

This law states that every pair of masses in the universe attracts each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It forms the basis for computing the gravitational force between large objects such as spacecraft and the ISS, and it uses the gravitational constant, masses, and distance in its calculation.

Inverse-Square Law

The inverse-square law emphasizes that as the distance between two masses increases, the gravitational force decreases rapidly—in proportion to the square of the distance. This concept is crucial when comparing the gravitational forces at different separations, such as when the Soyuz is docked versus when it is farther away from the ISS.

Thrust and Fuel Consumption

Thrust in rocket propulsion is generated by ejecting mass at high speed, and it is quantified by the equation F = ? × v?, where F is the thrust force, ? is the mass flow rate of the fuel, and v? is the exhaust velocity. This relationship allows one to calculate the necessary fuel consumption rate required to produce a specific thrust force that counteracts gravitational pull.

Average Force Calculation

When the force between objects varies with distance, determining the average force over a range involves integrating the variable force equation over that distance and then dividing by the interval. This approach is used to find an effective constant force that approximates the variable gravitational attraction for further calculations such as required thrust or fuel burn over time.

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