Answered: A new design of Saturn 19 rocket with…

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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Question

A new design of Saturn 19 rocket with mass 1.53 x 10⁶ kg has a propulsion engine that can eject gases at a constant speed of 392.1 m/s (vGas) by burning fuel at a rate of 2.77 x 10⁵ kg/s (dmdT). In full tank, the rocket can carry a fuel of mass 49.84 x 10⁶ kg. If this design exceeds the velocity of 11,200 m/s it can successfully escape Earth's gravity. Determine if this is so. Find its maximum velocity (velyZ). (Pitch Angle = 0)

Expert Solution

Step 1

Given:

Mass of rocket M = $1.53 \times 10^{6} k g$

Speed of ejection of gases $v_{g a s} = 392.1 m / s .$

Rate of burning of fuel $R = \frac{d m}{d t} = 2.77 \times 10^{5} k g / s .$

Mass of fuel m = $49.84 \times 10^{6} k g .$

Step 2

for the instantaneous mass the equation is,

$m \frac{d v}{d t} + v_{g a s} \frac{d m}{d t} = 0 m \frac{d v}{d t} = - v_{g a s} \frac{d m}{d t} d v = - v_{g a s} \frac{d m}{m} \int d v = - v_{g a s} \int_{m_{t o t a l}}^{M} \frac{d m}{m} v = v_{g a s} \ln (\frac{m_{t o t a l}}{M}) v = 392.1 \ln (\frac{(1.53 + 49.84) \times 10^{6}}{1.53 \times 10^{6}}) v = 392.1 \times 3.514 v = 1377.76 m / s$

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