Equations of motion for missiles under constant acceleration

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. I then figured out the appropriate set of Special Relativity equations that describe the coordinate time, coordinate speed and coordinate distance (aka: from the point of view of a ship those missiles are targeting) as a function of the proper time (aka: from of view of the missile under constant acceleration). These are the ones describing hyperbolic motion under Special Relativity:

https://en.wikipedia.org/wiki/Hyperboli ... relativity)

The Newtonian equations used in the Honorverse are:
u = u0+α*τ
X = u0*τ+α*τ^2/2
ΔT = τ

However, the hyperbolic motion (Special Relativity) equations are:
u(τ) = c*(tanh(atanh(u0/c)+α*τ/c))
ΔX(τ) = X(τ)-X0 = (c^2/α)*(cosh(atanh(u0/c)+α*τ/c)-c/sqrt(c^2-u0^2))
ΔT(τ) = T(τ)-T0 = (c/α)*(sinh(atanh(u0/c)+α*τ/c)-u0/sqrt(c^2-u0^2))

Where:
c is the speed of light (299792.458 km/s)
α is the proper acceleration experienced by the missile in km/s^2; I used 46,000 gravities for shipkillers and 130,000 gravities for CMs
τ is the proper time experienced by the missile in seconds
u0 is the initial speed speed of the missile
u (km/s), ΔX (km), and ΔT (s) are the speed at the end of run, range under power and time under power in Newtonian Physics
u(τ), (km/s), ΔX(τ) (km), and ΔT(τ) (s) are the speed at the end of run, range under power and time under power under Hyperbolic motion under Special Relativity. The (τ) symbol means they are calculated as a function of the proper time (τ) of the missile, and not as a function of the coordinate time (T) of the ship.

I made an Excel table comparing the results using Newtonian equations to those using Hyperbolic motion (Special Relativity) equations for several RMB CMs and shipkillers at rest and with a 0.1 c initial velocity relative to the enemy ships:

Motion variables Newtonian results
Missile α(km/s^2) τ(s) u0(km/s) u(km/s) ΔX(km) ΔT(s)
Mk-31 from rest 1275 75 0.0 95,625.0 3,585,937.5 75
Mk-31 from 0.1 c 1275 75 29,979.2 125,604.2 5,834,380.9 75
Mk-31 from 0.5 c 1275 75 149,896.2 245,521.2 14,828,154.7 75
Mk-13 from rest 451 180 0.0 81,180.0 7,306,200.0 180
Mk-13 from 0.1 c 451 180 29,979.2 111,159.2 12,702,464.2 180
Mk-16 from rest 451 360 0.0 162,360.0 29,224,800.0 360
Mk-16 from 0.1 c 451 360 29,979.2 192,339.2 40,017,328.5 360
Mk23 from rest 451 540 0.0 243,540.0 65,755,800.0 540
Mk-23 from 0.1 c 451 540 29,979.2 273,519.2 81,944,592.7 540
Mk-25 from rest 451 720 0.0 324,720.0 116,899,200.0 720
Mk-25 from 0.1 c 451 720 29,979.2 354,699.2 138,484,257.0 720


Motion variables Hyperbolic motion results
Missile α(km/s^2) τ(s) u0(km/s) u(τ)(km/s) ΔX(τ)(km) ΔT(τ)(s)
Mk-31 from rest 1275 75 0.0 92,508.7 3,616,444.3 76.3
Mk-31 from 0.1 c 1275 75 29,979.2 118,821.4 5,932,948.4 77.9
Mk-31 from 0.5 c 1275 75 149,896.2 210,003.9 17,378,553.0 95.0
Mk-13 from rest 451 180 0.0 79,252.3 7,350,953.7 182.2
Mk-13 from 0.1 c 451 180 29,979.2 106,418.3 12,877,959.2 185.6
Mk-16 from rest 451 360 0.0 148,151.1 29,946,130.8 377.9
Mk-16 from 0.1 c 451 360 29,979.2 169,742.1 41,481,961.9 389.8
Mk23 from rest 451 540 0.0 201,128.1 69,452,488.9 601.4
Mk-23 from 0.1 c 451 540 29,979.2 216,577.4 87,922,266.2 627.7
Mk-25 from rest 451 720 0.0 238,144.4 128,784,607.1 869.3
Mk-25 from 0.1 c 451 720 29,979.2 248,392.2 155,625,002.6 916.8

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.

Here is a table with the variations due to switching to Hyperbolic motion equations:

Percentage Hyperbolic/Newtonian
Missile u(τ)/u ΔX(τ)/ΔX ΔT(τ)/ΔT
Mk-31 from rest 96.74% 100.85% 101.70%
Mk-31 from 0.1 c 94.60% 101.69% 103.83%
Mk-31 from 0.5 c 85.53% 117.20% 126.72%
Mk-13 from rest 97.63% 100.61% 101.23%
Mk-13 from 0.1 c 95.74% 101.38% 103.11%
Mk-16 from rest 91.25% 102.47% 104.96%
Mk-16 from 0.1 c 88.25% 103.66% 108.28%
Mk23 from rest 82.59% 105.62% 111.37%
Mk-23 from 0.1 c 79.18% 107.29% 116.24%
Mk-25 from rest 73.34% 110.17% 120.73%
Mk-25 from 0.1 c 70.03% 112.38% 127.34%

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.

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. I then figured out the appropriate set of Special Relativity equations that describe the coordinate time, coordinate speed and coordinate distance (aka: from the point of view of a ship those missiles are targeting) as a function of the proper time (aka: from of view of the missile under constant acceleration). These are the ones describing hyperbolic motion under Special Relativity:

https://en.wikipedia.org/wiki/Hyperboli ... relativity)

The Newtonian equations used in the Honorverse are:
u = u0+α*τ
X = u0*τ+α*τ^2/2
ΔT = τ

However, the hyperbolic motion (Special Relativity) equations are:
u(τ) = c*(tanh(atanh(u0/c)+α*τ/c))
ΔX(τ) = X(τ)-X0 = (c^2/α)*(cosh(atanh(u0/c)+α*τ/c)-c/sqrt(c^2-u0^2))
ΔT(τ) = T(τ)-T0 = (c/α)*(sinh(atanh(u0/c)+α*τ/c)-u0/sqrt(c^2-u0^2))

Where:
c is the speed of light (299792.458 km/s)
α is the proper acceleration experienced by the missile in km/s^2; I used 46,000 gravities for shipkillers and 130,000 gravities for CMs
τ is the proper time experienced by the missile in seconds
u0 is the initial speed speed of the missile
u (km/s), ΔX (km), and ΔT (s) are the speed at the end of run, range under power and time under power in Newtonian Physics
u(τ), (km/s), ΔX(τ) (km), and ΔT(τ) (s) are the speed at the end of run, range under power and time under power under Hyperbolic motion under Special Relativity. The (τ) symbol means they are calculated as a function of the proper time (τ) of the missile, and not as a function of the coordinate time (T) of the ship.

I made an Excel table comparing the results using Newtonian equations to those using Hyperbolic motion (Special Relativity) equations for several RMB CMs and shipkillers at rest and with a 0.1 c initial velocity relative to the enemy ships:

Motion variables Newtonian results
Missile α(km/s^2) τ(s) u0(km/s) u(km/s) ΔX(km) ΔT(s)
Mk-31 from rest 1275 75 0.0 95,625.0 3,585,937.5 75
Mk-31 from 0.1 c 1275 75 29,979.2 125,604.2 5,834,380.9 75
Mk-31 from 0.5 c 1275 75 149,896.2 245,521.2 14,828,154.7 75
Mk-13 from rest 451 180 0.0 81,180.0 7,306,200.0 180
Mk-13 from 0.1 c 451 180 29,979.2 111,159.2 12,702,464.2 180
Mk-16 from rest 451 360 0.0 162,360.0 29,224,800.0 360
Mk-16 from 0.1 c 451 360 29,979.2 192,339.2 40,017,328.5 360
Mk23 from rest 451 540 0.0 243,540.0 65,755,800.0 540
Mk-23 from 0.1 c 451 540 29,979.2 273,519.2 81,944,592.7 540
Mk-25 from rest 451 720 0.0 324,720.0 116,899,200.0 720
Mk-25 from 0.1 c 451 720 29,979.2 354,699.2 138,484,257.0 720


Motion variables Hyperbolic motion results
Missile α(km/s^2) τ(s) u0(km/s) u(τ)(km/s) ΔX(τ)(km) ΔT(τ)(s)
Mk-31 from rest 1275 75 0.0 92,508.7 3,616,444.3 76.3
Mk-31 from 0.1 c 1275 75 29,979.2 118,821.4 5,932,948.4 77.9
Mk-31 from 0.5 c 1275 75 149,896.2 210,003.9 17,378,553.0 95.0
Mk-13 from rest 451 180 0.0 79,252.3 7,350,953.7 182.2
Mk-13 from 0.1 c 451 180 29,979.2 106,418.3 12,877,959.2 185.6
Mk-16 from rest 451 360 0.0 148,151.1 29,946,130.8 377.9
Mk-16 from 0.1 c 451 360 29,979.2 169,742.1 41,481,961.9 389.8
Mk23 from rest 451 540 0.0 201,128.1 69,452,488.9 601.4
Mk-23 from 0.1 c 451 540 29,979.2 216,577.4 87,922,266.2 627.7
Mk-25 from rest 451 720 0.0 238,144.4 128,784,607.1 869.3
Mk-25 from 0.1 c 451 720 29,979.2 248,392.2 155,625,002.6 916.8

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.

Here is a table with the variations due to switching to Hyperbolic motion equations:

Percentage Hyperbolic/Newtonian
Missile u(τ)/u ΔX(τ)/ΔX ΔT(τ)/ΔT
Mk-31 from rest 96.74% 100.85% 101.70%
Mk-31 from 0.1 c 94.60% 101.69% 103.83%
Mk-31 from 0.5 c 85.53% 117.20% 126.72%
Mk-13 from rest 97.63% 100.61% 101.23%
Mk-13 from 0.1 c 95.74% 101.38% 103.11%
Mk-16 from rest 91.25% 102.47% 104.96%
Mk-16 from 0.1 c 88.25% 103.66% 108.28%
Mk23 from rest 82.59% 105.62% 111.37%
Mk-23 from 0.1 c 79.18% 107.29% 116.24%
Mk-25 from rest 73.34% 110.17% 120.73%
Mk-25 from 0.1 c 70.03% 112.38% 127.34%

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.

Thank you, oh so verrrrry much for glazing my eyes over. All that stuff is pretty much what I skim over to get to the words I can understand.

My only teensy possible input is unless the missile pods are lying doggo, they start from the base velocity of the ship they are launched from + the acceleration from the launcher. At least if i recall my high school physics.

pappilon wrote:Thank you, oh so verrrrry much for glazing my eyes over. All that stuff is pretty much what I skim over to get to the words I can understand.

My only teensy possible input is unless the missile pods are lying doggo, they start from the base velocity of the ship they are launched from + the acceleration from the launcher. At least if i recall my high school physics.

Yes, that is true.
I added the some examples with initial velocity because in many battles the ships are described as moving towards each other at 5-15% the speed of light.

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. I then figured out the appropriate set of Special Relativity equations that describe the coordinate time, coordinate speed and coordinate distance (aka: from the point of view of a ship those missiles are targeting) as a function of the proper time (aka: from of view of the missile under constant acceleration). These are the ones describing hyperbolic motion under Special Relativity:

https://en.wikipedia.org/wiki/Hyperboli ... relativity)

The Newtonian equations used in the Honorverse are:
u = u0+α*τ
X = u0*τ+α*τ^2/2
ΔT = τ

However, the hyperbolic motion (Special Relativity) equations are:
u(τ) = c*(tanh(atanh(u0/c)+α*τ/c))
ΔX(τ) = X(τ)-X0 = (c^2/α)*(cosh(atanh(u0/c)+α*τ/c)-c/sqrt(c^2-u0^2))
ΔT(τ) = T(τ)-T0 = (c/α)*(sinh(atanh(u0/c)+α*τ/c)-u0/sqrt(c^2-u0^2))

Where:
c is the speed of light (299792.458 km/s)
α is the proper acceleration experienced by the missile in km/s^2; I used 46,000 gravities for shipkillers and 130,000 gravities for CMs
τ is the proper time experienced by the missile in seconds
u0 is the initial speed speed of the missile
u (km/s), ΔX (km), and ΔT (s) are the speed at the end of run, range under power and time under power in Newtonian Physics
u(τ), (km/s), ΔX(τ) (km), and ΔT(τ) (s) are the speed at the end of run, range under power and time under power under Hyperbolic motion under Special Relativity. The (τ) symbol means they are calculated as a function of the proper time (τ) of the missile, and not as a function of the coordinate time (T) of the ship.

I made an Excel table comparing the results using Newtonian equations to those using Hyperbolic motion (Special Relativity) equations for several RMB CMs and shipkillers at rest and with a 0.1 c initial velocity relative to the enemy ships:

Motion variables Newtonian results
Missile α(km/s^2) τ(s) u0(km/s) u(km/s) ΔX(km) ΔT(s)
Mk-31 from rest 1275 75 0.0 95,625.0 3,585,937.5 75
Mk-31 from 0.1 c 1275 75 29,979.2 125,604.2 5,834,380.9 75
Mk-31 from 0.5 c 1275 75 149,896.2 245,521.2 14,828,154.7 75
Mk-13 from rest 451 180 0.0 81,180.0 7,306,200.0 180
Mk-13 from 0.1 c 451 180 29,979.2 111,159.2 12,702,464.2 180
Mk-16 from rest 451 360 0.0 162,360.0 29,224,800.0 360
Mk-16 from 0.1 c 451 360 29,979.2 192,339.2 40,017,328.5 360
Mk23 from rest 451 540 0.0 243,540.0 65,755,800.0 540
Mk-23 from 0.1 c 451 540 29,979.2 273,519.2 81,944,592.7 540
Mk-25 from rest 451 720 0.0 324,720.0 116,899,200.0 720
Mk-25 from 0.1 c 451 720 29,979.2 354,699.2 138,484,257.0 720


Motion variables Hyperbolic motion results
Missile α(km/s^2) τ(s) u0(km/s) u(τ)(km/s) ΔX(τ)(km) ΔT(τ)(s)
Mk-31 from rest 1275 75 0.0 92,508.7 3,616,444.3 76.3
Mk-31 from 0.1 c 1275 75 29,979.2 118,821.4 5,932,948.4 77.9
Mk-31 from 0.5 c 1275 75 149,896.2 210,003.9 17,378,553.0 95.0
Mk-13 from rest 451 180 0.0 79,252.3 7,350,953.7 182.2
Mk-13 from 0.1 c 451 180 29,979.2 106,418.3 12,877,959.2 185.6
Mk-16 from rest 451 360 0.0 148,151.1 29,946,130.8 377.9
Mk-16 from 0.1 c 451 360 29,979.2 169,742.1 41,481,961.9 389.8
Mk23 from rest 451 540 0.0 201,128.1 69,452,488.9 601.4
Mk-23 from 0.1 c 451 540 29,979.2 216,577.4 87,922,266.2 627.7
Mk-25 from rest 451 720 0.0 238,144.4 128,784,607.1 869.3
Mk-25 from 0.1 c 451 720 29,979.2 248,392.2 155,625,002.6 916.8

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.

Here is a table with the variations due to switching to Hyperbolic motion equations:

Percentage Hyperbolic/Newtonian
Missile u(τ)/u ΔX(τ)/ΔX ΔT(τ)/ΔT
Mk-31 from rest 96.74% 100.85% 101.70%
Mk-31 from 0.1 c 94.60% 101.69% 103.83%
Mk-31 from 0.5 c 85.53% 117.20% 126.72%
Mk-13 from rest 97.63% 100.61% 101.23%
Mk-13 from 0.1 c 95.74% 101.38% 103.11%
Mk-16 from rest 91.25% 102.47% 104.96%
Mk-16 from 0.1 c 88.25% 103.66% 108.28%
Mk23 from rest 82.59% 105.62% 111.37%
Mk-23 from 0.1 c 79.18% 107.29% 116.24%
Mk-25 from rest 73.34% 110.17% 120.73%
Mk-25 from 0.1 c 70.03% 112.38% 127.34%

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.

I don't remember a MK 31 in the series - By Mk 31, do you mean the Mk 41 capacitor Multi drive missile, or another missile, Like the Mk 34 Destroyer weight missile or the Mk 36 LERM?

Theemile wrote:
I don't remember a MK 31 in the series - By Mk 31, do you mean the Mk 41 capacitor Multi drive missile, or another missile, Like the Mk 34 Destroyer weight missile or the Mk 36 LERM?

No, I was talking about the RMN Mark 31 counter-missile that first appeared in AAC:

"The Mark 31 counter-missiles Honor's ships were firing represented significant improvements even over the Mark 30 counter-missiles her command had used as recently as the Battle of Sidemore, only months before. Their insanely powerful wedges were capable of sustaining accelerations of up to 130,000 for as much as seventy-five seconds, which gave them a powered range from rest of almost 3.6 million kilometers."

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: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. I then figured out the appropriate set of Special Relativity equations that describe the coordinate time, coordinate speed and coordinate distance (aka: from the point of view of a ship those missiles are targeting) as a function of the proper time (aka: from of view of the missile under constant acceleration). These are the ones describing hyperbolic motion under Special Relativity:

There have been the occasional discussion over the years about Newtonian vs special relativity in ships and missiles.

I got deep into it a few years ago, ultimately leading to the spreadsheets I posted here: viewtopic.php?f=1&t=8126

I agree that Newtonian acceleration formulas give non-nonsensical answers in the face of MDMs. The problem is that (presumably to keep his math simple) David seems to exclusvely use them. Any time we get a full set of missile performance numbers, time, max distance, and terminal velocity they agree with Newton and not Einstein.

I suspect he'll end up throwing an arbitrary 0.9c cap on their velocity (to stay consistent with the relativistic bombardment statements from some of the early books) but otherwise leave them Newtonian until they reach that speed limit.

While I could wade through the math it's been long enough it wouldn't be easy.

I do get the impression that the corrected numbers are a bit on the low side, though--is the burn time figured in ship time or missile time???

Loren Pechtel wrote:While I could wade through the math it's been long enough it wouldn't be easy.

I do get the impression that the corrected numbers are a bit on the low side, though--is the burn time figured in ship time or missile time???

The numbers for the Mk-23 and -25 from rest are right; the burn times (2nd column) are 540 and 720 s respectively (9 and 12 min). That's missile time; from the ship, the observed burn times (last column) are 601 and 869 s (10 and 14.5 min).

Bill Woods wrote:

Loren Pechtel wrote:While I could wade through the math it's been long enough it wouldn't be easy.

I do get the impression that the corrected numbers are a bit on the low side, though--is the burn time figured in ship time or missile time???

The numbers for the Mk-23 and -25 from rest are right; the burn times (2nd column) are 540 and 720 s respectively (9 and 12 min). That's missile time; from the ship, the observed burn times (last column) are 601 and 869 s (10 and 14.5 min).

Exactly. The Missile 'sees' 540 or 720 s, but the target ship sees a longer time period (the missile moves at relativistic speed, so its time is somewhat contracted).