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["SPACESHIP INTERCEPTION - PURSUIT - THE GENERAL EQUATION"]
[]
["This is the solution of the Simultaneous Equations describing the pursuit and interception of one spaceship by another."]
["Situation:","Both Target and Pursuer start within the System M25. Both have the same trajectory i.e. moving along the same line."]
[null,"Both MAY possess an initial velocity with respect to the system."]
[null,"Velocity is positive in the direction of acceleration which is toward the M25."]
[null,"The Target MAY start accelerating some time before the Pursuer.  "]
[null,"The Target and Pursuer MAY start some distance apart."]
[null,"The Pursuer's acceleration must be greater than the Target's."]
[]
["Known","Initial","Initial","Time","Acceleration",null,"Calculated ","Velocity","Distance","Time to"]
["Data","Separation","Velocity","Delay",null,null,"Data",null,"from datum","Intercept"]
["Target  T","L","Ut","S","At",null,"Target  T","Vt","Dt","T"]
["Pursuer P",null,"Up",null,"Ap",null,"Pursuer P","Vp","Dp","T"]
["Units","length","speed","time","acceleration"]
[]
["Interception occurs when both the Target's and Pursuer's velocity and distance from the datum coincide.  This is achieved by the"]
["Pursuer accelerating for time 'x' until its velocity exceeds the Target's sufficiently and then decelerates for time 'y' until velocity"]
["and distance coincide.  Total time from when the Pursuer starts to accelerate until interception is 'T'.  T = x + y"]
[]
["EQUATIONS:",{"$nulls":7},"Equation No."]
["TARGET",null,"Vt = Ut + At.( T + S )",{"$nulls":5},"[ 1 ]"]
[]
[null,null,"Dt = L + Ut.( T + S ) + ( At/2 ).( T + S ) ^ 2",{"$nulls":5},"[ 2 ]"]
[]
["PURSUER",null,"Vp = Up + Ap.( T - y ) - Ap.y"]
[null,null,"Vp = Up + Ap.( T - 2.y )",{"$nulls":5},"[ 3 ]"]
[]
[null,null,"Dp = Up.T + ( Ap/2 ).( T - y ) ^ 2 + ( Ap/2 ).( 2.T.y - 3.y^2 )"]
[null,null,"Dp = Up.T + ( Ap/2 ).( T^2 + 2.y^2 )",{"$nulls":5},"[ 4 ]"]
[]
["AT INTERCEPTION",null,"Vt = Vp","and","Dt = Dp"]
["Combining the Velocity equations  [ 1 ] and [ 3 ]"]
[]
[null,null,"Ut + At.( T + S ) = Up + Ap.( T - 2.y )",{"$nulls":5},"[ 5 ]"]
[]
["Combining the Distance equations  [ 2 ] and [ 4 ]"]
[]
[null,"L + Ut.( T + S ) + ( At/2 ).( T + S )^2 = Up.T + ( Ap/2 ).( T^2 + 2.y^2 )",{"$nulls":6},"[ 6 ]"]
[]
["Rearranging [ 5 ] to make 'y' the subject:"]
[]
[null,"y = [ ( Up - Ut - At.S ) + ( Ap - At ).T ] / ( 2.Ap )"]
[]
["Substituting for 'y' in [ 6 ] and then rearranging in the form:    a.T^2 + b.T + c = 0"]
[]
[null,"L + Ut.T + Ut.S + ( At/2 ).( T^2 + 2.T.S + S^2 ) - Up.T - ( Ap/2 ).T^2 + Ap.y^2 = 0"]
[]
[null,"y ^ 2 = [ ( Up - Ut - At.S )^2 + 2.( Up - Ut - At.S ).( Ap - At ) + ( Ap - At )^2.T^2 ] / ( 4.Ap^2 )"]
[]
[null,"L + Ut.T + Ut.S + ( At/2 ).( T^2 + 2.T.S + S^2 ) - Up.T - ( Ap/2 ).T^2 + ( Up - Ut - At.S )^2 / ( 4.Ap^2 )"]
[null," +  2.( Up - Ut - At.S ).( Ap - At ).T / ( 4.Ap^2 ) + ( Ap - At )^2.T^2 ] / ( 4.Ap^2 ) = 0"]
[]
["\"a\"","[ At/2 - Ap/2 + ( Ap - At )^2 / ( 4.Ap ) ] . T^2 +"]
[]
["\"b\"","[ Ut + At.S - Up + ( Up - Ut - At.S ).( Ap - At ) / ( 2.Ap ) ] . T +"]
[]
["\"c\"","[ L + Ut.S + ( At/2 ).S^2 + ( Up - Ut - At.S )^2 / ( 4.Ap ) ]  = 0"]
[]
["Solution:",null,"T  =  [ - b +/- SQRT ( B^2 - 4.a.c ) ] / ( 2.a )"]
[]
[{"$nulls":11},"1 Au / Day = (m/s)",null,"1g per day in m/s",864000]
["Standard Data:",null,"1 Au = 150 x 10^9 metres",{"$nulls":8},1736111.111111111,null,"1 AU in meters",150000000000]
[null,null,"1.0 G = 10 metres per second per second"]
[]
["DATA","Target","Target ","Initial","Delay in","Pursuer","Pursuer","Pursuer","Time to","Target","Intercept"]
[null,"Acc.n","Starting","Distance","Pursuer","Acc.n","Initial","Distance","Intercept","Distance ","Success"]
[null,null,"Velocity","twixt ships","start",null,"Velocity","to M25",null,"to Intercept"]
["UNITS","G","Au / day","Au","hours","G","Au / day","Au","Days","Au","Y / N","a","b","c"]
[null,"At","Ut","L","S","Ap","Up"]
["Example",1,0,24,1,-1e-05,3.5,48,7.021967680295784,36.41545526843781,"GOTCHA!",-249999.999975,302016424249.99994,-9.121214475773083e+16]
[]
[{"$nulls":6},"Target velocity",3.494580523646721]
[{"$nulls":6},"Pursuer velocity",3.4999650541947633]

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