   Chapter 13.4, Problem 46E

Chapter
Section
Textbook Problem

# A rocket burning its onboard fuel while moving through space has velocity v(t) and mass m(t) at time t. If the exhaust gases escape with velocity ve relative to the rocket, it can be deduced from Newton’s Second Law of Motion that m d v d t = d m d t v e (a) Show that v ( t ) = v ( 0 ) − ln m ( 0 ) m ( t ) v e .(b) For the rocket to accelerate in a straight line from rest to twice the speed of its own exhaust gases, what fraction of its initial mass would the rocket have to bum as fuel?

(a)

To determine

To show: The vector v(t)=v(0)lnm(0)m(t)ve .

Explanation

Given data:

mdvdt=dmdtve (1)

Rearrange the equation (1).

dvdt=1mdmdtve

Take integration with respect to t on both sides.

0tdvdtdt=ve0t1mdmdtdtv(0)v(t)dv=vem(0)m(t)dm

(b)

To determine

To find: The fraction of its initial mass would have to burn the rocket as fuel.

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