You are the ground control commander of a 4000 kg scientific rocket that is appr

Question

You are the ground control commander of a 4000 kg scientific rocket that is approaching Mars at a speed of 22227 km/hr. It needs to quickly slow to 15000 km/hr to begin a controlled descent to the surface. If the rocket enters the Martian atmosphere too fast it will burn up, and if it enters too slowly, it will use up its maneuvering fuel before reaching the surface and will crash. The rocket has a new braking system: Several 5.0 kg "bullets" on the front of the rocket can be fired straight ahead. Each has a high-explosive charge that fires it at a speed of 146000 m/s relative to the rocket. You need to send the rocket an instruction to tell it how many bullets to fire. Success will bring you fame and glory, but failure of this $500,000,000 mission will ruin your career. How many bullets do you need to fire?

Explanation / Answer

initial speed of the rocket=22227 km/hr=22227*1000/3600 m/s

=6174.2 m/s

final speed of the rocket=15000 km/hr=15000*1000/3600 m/s

=4166.7 m/s

let number of bullets to be fired is N.

speed of the bullet , going forward with respect to an observer at rest=relative speed with repsect to rocket+speed of the rocket

=146000+6174.2

=1.5217*10^5 m/s

then using conservation of momentum principle:

initial total momentum=final total momentum

==>4000*6174.2=(4000-5*N)*4166.7+5*N*1.5217*10^5

==>4000*6175.2-4000*4166.7=5*N*(1.5217*10^5-4166.7)

==>N=(4000*6175.2-4000*4166.7)/(5*(1.5217*10^5-4166.7))

==>N=10.857

hence you should fire in total 11 bullets.

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