. Simple satellite / rocket seperation problem is my thinking wrong?

[Tschew]

URGENT. Simple satellite / rocket separation problem... is my thinking wrong?

Rocket mass M-kM and satellite kM (k < 0) are separated by an explosion which releases energy Q and let's rocket M-kM come to a stop relative to the observer and satellite kM continue at velocity v. Initial velocity u.

One has to show that v² = 2Q / kM(1-k) the solution of which is rather trivial, when using the following relationship:

(1) 1/2 (M-kM) u² - Q = 0

and then

(2) 1/2 kM u² + Q = 1/2 kM v²

replacing u² from (1)

However, first of all, if all energy is used up to decelerate the rocket, where does the extra kinetic energy of the satellite come from? Shouldn't half the energy go to the rocket and half to the satellite? (Of course, all this considering an instantaneous transfer of energy)

Also, since momentum should be conserved then shouldn't this be true?

M u = kM v [since M-kM comes to a halt]

Now, replacing everything with numbers:

M=5 k=0.2 kM=1 u = 3

1/2*(4)*9 = Q = 18

1/2*1*9 + 18 = 1/2*1*v²

9+36=v²

45 = v²

v = 6.7

--> 5 * 3 = 1 * 6.7 ? NOPE

Also: v² = 2Q / kM(1-k) = 36 / 0.8 = 45 which is certainly the same as otherwise, but still momentum is not conserved.. although it should be right?

Wouldn't it be much more useful to say:

M*u = kM*v

v = u/k ?

Please clarify the issue as I seem to have serious problems understanding it even though it is a simple linear momentum / energy problem! grr!

Thanks.

 

[Doc Al]

One has to show that v² = 2Q / kM(1-k) the solution of which is rather trivial, when using the following relationship:

(1) 1/2 (M-kM) u² - Q = 0

and then

(2) 1/2 kM u² + Q = 1/2 kM v²

replacing u² from (1)

I'm not sure where you get those equations, but they are incorrect. Note that if you add them, they imply that the KE of the system did not change! What happened to Q?

Also, since momentum should be conserved then shouldn't this be true?

M u = kM v [since M-kM comes to a halt]

Absolutely. Start with this equation for momentum conservation, then add the fact that the KE of the entire system is increased by Q. (Make the assumption that all the energy of the explosion goes into the mechanical energy of the system.)

 

[wais]

It seems like you have a good understanding of the problem and the equations involved. However, there are a few things that may be causing confusion. First, it's important to note that the explosion is not instantaneous and there is a period of time where the energy is being transferred from the explosion to the rocket and satellite. This means that the rocket and satellite will not come to a complete stop and continue at a constant velocity, but rather they will experience a change in velocity over time.

Secondly, the energy Q is not solely used to decelerate the rocket. Some of the energy will also go towards accelerating the satellite, which is why the kinetic energy of the satellite is not equal to the kinetic energy of the rocket. This is also why the momentum is not conserved in your calculations.

Finally, your equation M*u = kM*v is not correct. This equation assumes that the rocket and satellite have the same initial velocity, which is not the case in this problem. Instead, the correct equation would be M*u = (M-kM)*v, which takes into account the fact that the rocket has a different initial velocity than the satellite.

I hope this helps clarify the issue for you. Keep in mind that these types of problems can be tricky and it's important to carefully consider all the variables and assumptions involved. Good luck!