Equations of motion for missiles under constant acceleration

Bill Woods wrote:

Hegemon wrote:Hello,

My search to previous posts on this topic have found nothing, so here it goes.

I knew that calculating the range and speed at the end of the run of impeller drive missiles using Newtonian Physics induces some errors compared to using Special Relativity. ...

[snip]
As you can see, the Newtonian equations break up completely on a four-stage missile like Mk-25, resulting in a speed at the end of run higher than the speed of light. The Hyperbolic motion equations do not have this problem.

[snip]
As you can see, the Hyperbolic motion equations always give lower values for velocity at the end of the run but higher range under power and time under power than the Newtonian equations.

I hope you found this interesting. Please tell me what you think.

A while back, I posted this:
viewtopic.php?f=1&t=7150&hilit=cheatsheet&start=32

[Edit] See also Jonathan's and my posts in
viewtopic.php?f=1&t=8922

Hegemon wrote:
Sorry, I did not find your posts. I searched specifically for hyperbolic motion. You indeed used the hyperbolic motion equations and your results are identical to mine, but did not explicitly mentioned it.

Anyway, I can say I added the correction needed to account for the initial velocity (a ship moving towards or away from the missile at launch).

Daryl wrote:An interesting topic, and one I have touched upon myself, although I'm too lazy to relearn the maths.
Just to go a step further, in the more extreme examples what would the final mass of the missile be? Plus would that make the kinetic energy ridiculously high? I'm thinking about the free energy supplied by Honorverse physics converted to additional mass at near C velocity, then expended in a collision with a planet for example.

Louis R wrote:I think that if you could actually test your results you'd find serious errors remain.

Special Relativity deals with _uniform_ motion. Including acceleration moves you into General Relativity - and even ships routinely use accels found in nature only in the immediate vicinity of black holes. With all that that implies for mass, energy and time.

Bill Woods wrote:

Daryl wrote:An interesting topic, and one I have touched upon myself, although I'm too lazy to relearn the maths.
Just to go a step further, in the more extreme examples what would the final mass of the missile be? Plus would that make the kinetic energy ridiculously high? I'm thinking about the free energy supplied by Honorverse physics converted to additional mass at near C velocity, then expended in a collision with a planet for example.

KE = (gamma-1) m c^2

After a full burn, a (relativistic) Mk-23 is going 0.67c, which gives it 7.5 megatons per kilogram of kinetic energy. (Put another way, any speck of dust in the missile's way will also pack a punch.)

Bill Woods wrote:

Daryl wrote:An interesting topic, and one I have touched upon myself, although I'm too lazy to relearn the maths.
Just to go a step further, in the more extreme examples what would the final mass of the missile be? Plus would that make the kinetic energy ridiculously high? I'm thinking about the free energy supplied by Honorverse physics converted to additional mass at near C velocity, then expended in a collision with a planet for example.

KE = (gamma-1) m c^2

After a full burn, a (relativistic) Mk-23 is going 0.67c, which gives it 7.5 megatons per kilogram of kinetic energy. (Put another way, any speck of dust in the missile's way will also pack a punch.)

Daryl wrote:What I was looking for was the over 0.9C scenaro where the mass of the missile has increased, due to relativistic effects. So you have the double whammy of enormous kinetic energy just from going so fast, added to by the actual mass increasing in a MC2=E situation.