/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 177 A small rocket motor is used to ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 177

A small rocket motor is used to power a "jet pack" device to lift a single astronaut above the Moon's surface. The rocket motor produces a uniform exhaust jet with constant specd, \(V_{e}=3000 \mathrm{m} / \mathrm{s},\) and the thrust is varied by changing the jet size. The total initial mass of the astronaut and the jet pack is \(M_{0}=200 \mathrm{kg}, 100 \mathrm{kg}\) of which is fuel and oxygen for the rocket motor. Find (a) the exhaust mass flow rate required to just lift off initially, (b) the mass flow rate just as the fuel and oxygen are used up, and (c) the maximum anticipated time of flight. Note that the Moon's gravity is about 17 percent of Earth's.

Short Answer

Expert verified
The exhaust mass flow rate required for takeoff is 0.111 kg/s, the mass flow rate when the fuel is used up is 0.0555 kg/s, and the maximum anticipated time of flight is 900 seconds or 15 minutes.

Step by step solution

01

Calculate the gravitational force on the Moon

We first need to calculate the gravitational force that acts on the astronaut on the Moon. This will allow us to find the force that needs to be overcome for the astronaut to lift off. The weight of an object is its mass times the gravitational acceleration, which on the Moon is 0.17 times the gravitational acceleration on Earth. Therefore, the weight of the astronaut and the jet pack on the Moon is \((200 \mathrm{kg} \cdot (9.8 m/s虏 \cdot 0.17) = 332.6 N\).
02

Calculate the mass flow rate for takeoff

The thrust needed for takeoff is equal to the weight of the astronaut and the jet pack. The thrust produced by a rocket is equal to the exhaust speed times the mass flow rate. Therefore, we can solve for the mass flow rate by dividing the thrust by the exhaust speed. The mass flow rate for takeoff is therefore \( (332.6 N / 3000 m/s) = 0.111 kg/s\).
03

Calculate the mass flow rate after fuel is gone

After the fuel is used up, the total mass of the astronaut and the jet pack is reduced by the mass of the fuel. The weight is therefore \( (100 kg \cdot (9.8 m/s虏 \cdot 0.17) = 166.6 N\). The mass flow rate at this point, assuming the exhaust speed remains constant, is therefore \((166.6 N / 3000 m/s = 0.0555 kg/s\).
04

Calculate the maximum time of flight

To find the maximum time of flight, we need to calculate how long it takes for all the fuel to be used up. This is done by dividing the total mass of the fuel by the initial mass flow rate. Therefore, the maximum time of flight is \((100 kg / 0.111 kg/s) = 900 s \, or \, 15 minutes.\)

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