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Space physics for rockets, spaceships - Learning Calculus edition

Suppose we have a rocket A with a position vector and velocity (excluding acceleration, as in many games) and a spaceship B with position and velocity vectors. Now this rocket, the Breakthrough Distortion Cancer, will try to find the best interception for Spacecraft B.

Rocket A has two advantages: it knows the calculus and can calculate the roots of polynomials. However, the rocket, or abstract, the programmer, is still studying calculus and wants to know if it has the correct equation. (Polynomial roots will be solved by a nice guy named Jenkins-Traub Code, implemented from Netlib)

In particular:

  • mp = Rocket Position

  • mv = rocket speed

  • sp = Spacecraft position

  • sv = Spacecraft speed

  • t = time

According to the best assumption of the programmer, the equation for interception is: tspsv + tspmv - tmpsv - tmpmv

In addition, I am sure that I am completely on the wrong track, as there must be some exponents in this mess; this is an attempt to solve: (SP-MP) (SV-mv) (t)

My other option distinguishes between (sp-mp) (sv-mv) ^ 2, but I wanted to get feedback first, partly because if I'm not mistaken, '(sp-mp)' allows '1', And it seems ... odd. OTOH, the speed with which this function changes may be what I'm looking for.

So - What did I get wrong, where and why?

Thank.

Potentially useful link to the first stream.

Edit:

Summing up the equations:

(a + bx) + (c + ex)

(a + 1bx ^ 0) + (c + 1ex ^ 0)

(a + 1) + (c + 1)

unviable.

Product Equations:

(a + bx) (c + ex)

ac + aeh + CBX + BEx ^ 2

( Jenkins-Traub) .

+ 1aex ^ 0 + 1cbx ^ 0 + 2bex ^ 1

+ + + CB 2bex

, .

+5
3

(, t = 0)

[ mpx(t) = mpx(0) + mvx*t , mpy(t) = mpy(0) + mvy*t ]


[ spx(t) = spx(0) + svx*t , spy(t) = spy(0) + svy*t ]

mpx(0) mpy(0) spx(0) spy(0) -

, , mpx(t)=spx(t) mpy(t)=spy(t). . , t, - , slope=mvy/mvx. mpx(0) mpy(0), .

, .

+2

Position_Ship + t*Velocity_Ship = Position_Missile + t*Velocity_Missile

, t .

Velocity_Missile, .

N = (Position_Missile - Position_Ship) ^ Velocity_Ship (-)

N dot Velocity_Missle = 0

.

Velocity_Missile , , , . , , ...

Velocity_Missile' - , t', , t'', . t''*Ship_Velocity, . .

0

mp, mv sp, sv t:

mp+mv(t)=sp+sv(t) and |sv|=q (maxspeed)
mp+mv(t)-sp+sv(t)=0

mpx+mvx*t-spx+svx*t=0
mpy+mvy*t-spy+svy*t=0

svx^2+svy^2=q^2
svx^2+svy^2-q^2=0

:

(mpx-spx)/t+mvx=svx
(mpy-spy)/t+mvy=svy

((mpx-spx)/t+mvx)^2+((mpy-spy)/t+mvy)^2=q^2
(mpx-spx)^2/t^2+2*mvx*(mpx-spx)/t+mvx^2+(mpy-spy)^2/t^2+2*mvy*(mpy-spy)/t+mvy^2=q^2
((mpx-spx)^2+(mpy-spy)^2)/t^2+(2*mvx*(mpx-spx)+2*mvy*(mpy-spy))/t+mvx^2+mvy^2-q^2=0
((mpx-spx)^2+(mpy-spy)^2)+(2*mvx*(mpx-spx)+2*mvy*(mpy-spy))*t+(mvx^2+mvy^2-q^2)*t^2=0
((mpx-spx)^2+(mpy-spy)^2)/(mvx^2+mvy^2-q^2)+(2*mvx*(mpx-spx)+2*mvy*(mpy-spy))/(mvx^2+mvy^2-q^2)*t+t^2=0

c = (mvx^2+mvy^2-q^2)
if   a = (2*mvx*(mpx-spx)+2*mvy*(mpy-spy))/c
and  b = ((mpx-spx)^2+(mpy-spy)^2)/c
then t = -a/2+-sqrt(a^2/4-b)

a/2 = (2*mvx*(mpx-spx)+2*mvy*(mpy-spy))/2c
a/2 = (mvx*(mpx-spx)+mvy*(mpy-spy))/c
a^2/4 = (mvx^2*(mpx-spx)^2+2*mvx*(mpx-spx)*mvy*(mpy-spy)+mvy^2*(mpy-spy)^2)/c^2

b/c^2=((mpx-spx)^2+(mpy-spy)^2)*c
b/c^2=((mpx-spx)^2+(mpy-spy)^2)*(mvx^2+mvy^2-q^2)
b/c^2=mvx^2(mpx-spx)^2+mvx^2(mpy-spy)^2+mvy^2(mpx-spx)^2+mvy^2(mpy-spy)^2-q^2(mpx-spx)^2-q^2(mpy-spy)^2

t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt((mvx^2*(mpx-spx)^2+2*mvx*(mpx-spx)*mvy*(mpy-spy)+mvy^2*(mpy-spy)^2)-(mvx^2(mpx-spx)^2+mvx^2(mpy-spy)^2+mvy^2(mpx-spx)^2+mvy^2(mpy-spy)^2-q^2(mpx-spx)^2-q^2(mpy-spy)^2))/c

t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt((2*mvx*(mpx-spx)*mvy*(mpy-spy))-(mvx^2(mpy-spy)^2+mvy^2(mpx-spx)^2)+q^2((mpx-spx)^2+(mpy-spy)^2)))/c

t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt(-(mvx(mpy-spy)-mvy(mpx-spx))^2+q^2((mpx-spx)^2+(mpy-spy)^2)))/c

t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt(-(mvx(mpy-spy)-mvy(mpx-spx))^2+q^2(c+q^2))/c
t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt(-(mvx(mpy-spy)-mvy(mpx-spx))^2+q^2(c+q^2))/c

t = -(mvx*(mpx-spx)+mvy*(mpy-spy))/c +- sqrt(-(mvx(mpy-spy)-mvy(mpx-spx))^2+q^2(c+q^2))/(mvx^2+mvy^2-q^2)

, .

:

svx=(mpx-spx)/t+mvx
svy=(mpy-spy)/t+mvy

s.

, - ...

0

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