Obviously, we can make rockets to go very fast, but what is a reasonable top spe
ID: 1518440 • Letter: O
Question
Obviously, we can make rockets to go very fast, but what is a reasonable top speed? Assume that a rocket is fired from rest at a space station in deep space, where gravity is negligible. Assume that c = 3.00 × 108 m/s .
A)If the rocket ejects gas at a relative speed of 1400 m/s and you want the rocket's speed eventually to be 2.00×10?3 c , where c is the speed of light, what fraction of the initial mass of the rocket and fuel is not fuel? 3 sig figs
B)What is this fraction if the final speed is to be 3200 m/s ? 3 sig figs
Explanation / Answer
Solution:
Answers: A) 7.48 x 10-187
B) 0.102 ( 0r 10.2% )
Velocity = 2x10^-3 c =6 x10^5 m/s
The relation between the mass variation and velocity is given as
v - vo = v exhaust ln (mo/m) where vo and mo are the initial velocity and mass .
6x10^5 - 0 = (1300) ln ( m/mo)
=> ln (mo/m) = 6 x10^5 /1400
= 428.57
=> m /mo = e^428.57 = 1.337 x 10^186
=> m/mo = 1 / 1.337 x 10^186 = 7.48 x 10-187
fraction of the initial mass of the rocket and fuel is not fuel = 7.48 x 10-187 .
B) ln (mo/m) = v/vexhaust = 3200 / 1400
=> ln ( mo/m) = 2.2857
=> mo / m = e^2.2867 = 9.83
=> m = mo /9.83 = 0.102
=>m =(10.2 % ) mo
=> 10.2% of the initial mass is not fuel.
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