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Solid rocket velocity and distance

  • #1
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Hi there. I'm new to the forum. I apologize if I'm posting at the wrong session. Anyway, here goes the problem: (sorry for any grammar typos).

A solid fuel rocket, home constructed, has initial mass 10 kg; this, fuel is 8 kg. The rocket is launched vertically, from rest; burning the fuel at a constant rate equal to 0.225 kg/s, ejecting the exhaustion gases at a speed of 1980 m/s in relation to the rocket. Assume that the outlet pressure is the atmospheric and that the air resistance can be neglected. Calculate the velocity of the rocket 20 seconds after the launch and the distance traveled in the same interval.

I tried to find the velocity through the integral equation of the conservation of mass, but I do not have the areas. I could assume that they simplify, although I am not sure how to deal with the specific mass or volume, since they vary. I imagine that the values will come by through some ODE.

Thanks, in advance.
 

Answers and Replies

  • #2
Ray Vickson
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Hi there. I'm new to the forum. I apologize if I'm posting at the wrong session. Anyway, here goes the problem: (sorry for any grammar typos).

A solid fuel rocket, home constructed, has initial mass 10 kg; this, fuel is 8 kg. The rocket is launched vertically, from rest; burning the fuel at a constant rate equal to 0.225 kg/s, ejecting the exhaustion gases at a speed of 1980 m/s in relation to the rocket. Assume that the outlet pressure is the atmospheric and that the air resistance can be neglected. Calculate the velocity of the rocket 20 seconds after the launch and the distance traveled in the same interval.

I tried to find the velocity through the integral equation of the conservation of mass, but I do not have the areas. I could assume that they simplify, although I am not sure how to deal with the specific mass or volume, since they vary. I imagine that the values will come by through some ODE.

Thanks, in advance.
There are numerous papers and articles about such "variable mass" dynamical systems; in particular, Google 'rocket equation'.
 
  • #3
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There are numerous papers and articles about such "variable mass" dynamical systems; in particular, Google 'rocket equation'.
Yes. I googled for that. Although they all deal with force and that's not the case of the problem. Or at least, that's what I guess. As I said above, I think it is only a problem of conservation of mass, not momentum. Nevertheless, I will give a second look.
 
  • #4
SteamKing
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Yes. I googled for that. Although they all deal with force and that's not the case of the problem. Or at least, that's what I guess. As I said above, I think it is only a problem of conservation of mass, not momentum. Nevertheless, I will give a second look.
If the rocket is burning fuel and ejecting the exhaust out the back, how can this be a conservation of mass?
 
  • #5
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If the rocket is burning fuel and ejecting the exhaust out the back, how can this be a conservation of mass?
You are right. My mistake. I thought the problem could be simplified only to a conservation of mass.

I searched for some equations on the internet and I solved like this:

gif.latex?R%20%3D%20%5Cfrac%7Bdm%7D%7Bdt%7D.gif


if.latex?%5Cint_%7Bt_%7B0%7D%7D%5E%7Bt%7D%20Rdt%20%5Cto%20m%28t%29%20%3D%20m_%7B0%7D%20-%20Rt%5C.gif


where "m0" is (rocket material + rocket fuel) and "R" is the constant rate of exhaustion. So, m0 = 10 kg and R = 0.225 kg/s. This yields m(t) = 5.5 kg.

Applying Tsiolkovsky rocket equation:

gif.latex?m%5Cfrac%7Bdv%7D%7Bdt%7D%20%3D%20-v_%7Brel%7D%5Cfrac%7Bdm%7D%7Bdt%7D.gif


gif.latex?v%20%3D%20-v_%7Brel%7Dln%5Cleft%20%28%20%5Cfrac%7Bm_%7B0%7D%7D%7Bm%7D%20%5Cright%20%29.gif


So:

0kg%7D%7B5%2C5%20%5C%3Bkg%7D%20%5Cright%20%29%20%5C%5C%20%5C%5C%20v%20%3D%20-1183.71%20%28%3F%29.gif
(m/s)

But this yields a negative velocity (not forgetting to mention that the rocket is displacing itself at a rate of 1.18 km/s). What have I done wrong?
 

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