I can't comprehend impulse = momentum

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The discussion revolves around the relationship between impulse and momentum in a rocket's motion. A calculation is presented showing that a force of 50N applied over a time period results in a specific acceleration and distance covered. There is confusion regarding whether the thrust must be applied for the entire duration to cover 1000m in approximately 204.98 seconds. It is clarified that the assumption of constant thrust throughout the entire travel time may be incorrect, suggesting that the thrust duration could vary. The conversation concludes with an acknowledgment of the initial misunderstanding and a newfound clarity on the problem.
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Homework Statement
A satellite with a mass of 1050kg without any momentum or influence of force is floating freely in space. Later, its rocket was activated, applying force of 50N for x seconds. The satellite travelled 1000m in y seconds. Find the time needed, x, to apply 50N.

So i used the equation f = ma to find a so that i can plug it in the equation s = ut + ½at ²
Relevant Equations
F = ma
S = ut + ½at ²
Here is my calculation:

F = ma
50N = 1050kg * a
a = 0.0476m/s²

S = ut + ½at ²
1000m = 0t + ½(0.0476)t²
t = 204.980s
y = 204.980s (time to travel 1000m)

since impulse = momentum,

F * t = mv
F * x = m * distance covered/y
50N * x = 1050kg * 1000m/204.980s
50N * x = 5122.450N⋅s
x = 102.440s

50N * 102.440s ≈ 1050kg * 4.88m/s

Does this mean that the rocket has to apply thrust of 50N for 102.440s to supply the speed needed to cover 1000m in 204.98s ?
 
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The wording is slightly confusing to me. Is the time interval ##y## measured after the burn has completed? Otherwise it seems to me that there are infinitely many solutions to this question.

(N.B. Also a tiny quibble from me, the impulse ##I## equals the change in momentum ##\Delta P##. But in your case, if the initial momentum is 0, the impulse is the final momentum.)
 
Pushies said:
Here is my calculation:

F = ma
50N = 1050kg * a
a = 0.0476m/s²

S = ut + ½at ²
1000m = 0t + ½(0.0476)t²
t = 204.980s
y = 204.980s (time to travel 1000m)
You are making the assumption that the rocket is activated for all the time it takes to reach 1000 m, which I think is incorrect.

The way I read the problem, you have two unknowns, x and y. What you will then get is x as a function of y. (Even if the rocket is activated for 1 s, it will eventually travel a distance of 1000 m, taking longer than of the rocket is activated for a longer time.)
 
Sorry for the confusion guys. Iwas tired at the moment of writing, now i can solve it after some time of contemplating. Thanks
 
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