Momentum or Kinetic Energy to find Acceleration?

[God Plays Dice]

Hi all,

I have charged particles (protons) accelerated by an electric field. If I add up all the momentum and find the change in momentum per sec I can find the acceleration of my craft by
dp/dt /M = a

If I add up all the kinetic energy of the particles and find the KE per sec, I can find the acceleration by
Sqrt (KE/t /0.5*M) = a

They are both similar calcs but give wildly different values. KE calc is 10^9 bigger.

So which is it? KE or p to find acceleration?

 

[gleem]

If I add up all the kinetic energy of the particles and find the KE per sec, I can find the acceleration by
Sqrt (KE/t /0.5*M) = a

This is not correct

 

[God Plays Dice]

Gleem can you elaborate please.

 

[God Plays Dice]

Is it more like
Sqrt(KE/0.5*M) / t = a

 

[gleem]

Power = dK.E./dt = Fv= mav

a = (dK.E./dt)/p

 

[God Plays Dice]

cheers, its in the right ballpark now

 

[God Plays Dice]

The only problem now is that I have a system with a ridiculously large power requirement, 5E+14W, with not a very large final acceleration of 23m/s^2 of a 100 ton craft. Something is still wrong

 

[God Plays Dice]

No wait, problem still exists, I didnt put brackets in

 

[God Plays Dice]

The KE accln is actually 10^4 over the momentum accln

 

[gleem]

So you are producing a force of 2.3E⁶Nt assuming the mass is 100 metric tons ie. 10⁵Kgs and needing 5E+14 watts you get a velocity greater than that of light!? did you consider that the protons might be relativistic? What is the accelerators potential difference?

 

[God Plays Dice]

I have used relativistic equations. I have accelerated protons to 0.99c. Do the mathematics, find momentum and KE (relativistic) of these then you will see that when you try to get acceleration or velocity even from these two they differ greatly.

 

[God Plays Dice]

The momentum values seem more real, I don't know where that huge KE comes from, gamma is only around 7.

 

[God Plays Dice]

In fact the discrepancy is there even with slow moving protons, the KE derived values are far greater than those taken from p.

 

[gleem]

So how did you calculate the thrust?

 

[God Plays Dice]

I haven't calcd thrust. Just taken the momentum or KE of the particles and assumed that is the same on the craft

 

[God Plays Dice]

Surely that must be right for momentum at least, straight conservation of momentum

 

[God Plays Dice]

I suppose I did calc thrust, when I found momentum delivered to particles per sec

 

[gleem]

But the momentum is constantly changing. So you need to consider the rate of change of momentum and not just the momentum.

the proper way to solve the rocket problem is to say that the force ie thrust is the rate of change of momentum which in this case is the rate at which the mass of protons is being expelled times their velocity. and this is equal to the rate of change of the momentum of the rocket.

v_protonsdm_protons/dt = M_rocketa

 

[God Plays Dice]

I got that. Conservation of momentum. However I also found the KE of the particles, which seems very very high, and found the acceleration from that Power and it turns out a totally different value to the one from momentum. Its way higher. So I have two questions, or really just one. Why are they different and why is the KE so high, its 10^14W! This serves a momentum that can only move 100 ton at 22m/s^2.

The saturn 5 rocket was 10^11W and was 3000 ton.

 

[gleem]

KE is measure in joules and power is measured in Joules/sec. You need a time (of burn) to convert one to the other

 

[God Plays Dice]

Well i brought the proton speed down from 0.99c to around 3000m/s, seems to have done the trick I not stuck with an impossible power requirement now

 

[gleem]

That should not have been a problem for now you are non relativistic. So maybe your relativistic approach was the problem.

 

[God Plays Dice]

I did the calc in Newtonian. Still same result KE differs from momentum

 

[gleem]

I did the calc in Newtonian. Still same result KE differs from momentum

I don't understand this statement?

 

[gleem]

Are you assuming that the acceleration is constant because the power is constant? It is not. As the vehicle speeds up the acceleration decreases.

 

[God Plays Dice]

All I'm saying is that the momentum and kinetic energy lead to different velocity/acceleration values

 

[nasu]

I haven't calcd thrust. Just taken the momentum or KE of the particles and assumed that is the same on the craft

The second part is not necessarily true. The internal forces can change the total KE. You cannot use conservation of KE. You have something like a reversed inelastic
collision. Your discrepancy shows that the KE is not conserved, as expected.

 

[God Plays Dice]

I lowered the speed of the exhaust particles and guess what? I can get the KE velocity less than the momentum velocity.

 

[jbriggs444]

I lowered the speed of the exhaust particles and guess what? I can get the KE velocity less than the momentum velocity.

Right. Which means that you are using the wrong formula. Can you explain in careful detail how you derived the "kinetic energy" formula that you are using? Are you, for instance, assuming the KE of the craft = KE of the ejected protons?

 

[God Plays Dice]

Yes. I take the sum of all the particles in terms of momentum, ie their mass X vel and then divide thru by mass of the craft. I'm sure this is correct. Then I take the sum of all the particles 0.5 mass X vel^2. And rearrange this equation to find vel of the craft plugging in mass of craft. When the vel of the particles is very high then the KE vel is much higher than the momentum vel. When the vel of the particles is quite slow I get a KE vel slower than momentum vel. Do the excel sheet yourself it takes only a couple of minutes just plug any mass values in and do a range of particle velocities youl get non compliant KE vel with p vel.