Solve Momentum Problem: Car w/ Gun & Man Fire Shells at 200 m/s

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The problem involves a car with a gun and a man on a frictionless surface, where the man fires shells at a speed of 200 m/s, causing the car to recoil. The discussion focuses on calculating the car's speed after the second shell is fired using conservation of momentum principles. The calculations show that the car's velocity can be determined by analyzing the system as a whole or by considering parts of it separately. There is a question raised about the justification for using different systems in a single problem and whether matching results indicate a deeper understanding of the physics involved. The conclusion emphasizes the importance of consistent results in physics problems to validate the applied methods.
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Homework Statement



A car with a gun and a man is rest on a frictionless floor.The total mass is 50m where m is the mass of a single shell.Now,the man fires each shell with a muzzle speed 200 m/s.and the car recoils.What is the speed aftyer the 2nd time firing?

Homework Equations



v_sc=v_sf-v_cf where v_sc,v_sf,
v_cf are velocity of shell w.r.t. car,velocity of shell w.r.t. floor, velocity of car w.r.t. floor

The Attempt at a Solution



R_CM=[1/(50m)][49m*r_cf+m*r_sf]

(d/dt)R_CM=[1/(50m)] [49m*v_cf+m* v_sf ]=0

49m*v_cf+m*[ v_sc + v_cf ] =0 from relevant eqn.

This gives,v_cf=-[v_sc/50 m/s
=-200/50 (i) m/s where (i) is the unit vector

v_sf=(49/50)v_sc

w.r.t the same frame,
{48m*v'_cf+m*[v'_sc+v'_cf]}+m*(49/50)v_sc=0
49v'_cf=-v'_sc-(49/50)v_sc
v'_cf=-200(1/50+1/49)(i)


Please check if I went wrong anywhere
 
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I think I am correct.However,I have noticed that even if you do not take the whole of the system in the second case,even then the problem can be done.Just we have to use a more familiar version of conservation of linear momentum and apply it over the region of interest.

I want to know to what extent it is justified to use two different systems in a SINGLE problem to have a unique result?MY intuition suggests when the results match there should be some deeper way of understanding the physics.
If the result does not match,...then...it is an accident?
 

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