Quadrilateral Speed - time Curve
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Quadrilateral speed time curve
Let
α = Acceleration in km per hour per second
βC = Coasting retardation in km per hour per second
β = Braking retardation in km per hour per second
V1 = Maximum speed at the end of acceleration ( km per hour )
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V2 = Speed at the end of coasting period ( km per hour )
T = Total time of run ( second )
Acceleration time in second t1 = V1 / α …. ( 1 )
Coasting time in second t2 = V1 – V2 / βC ….. ( 2 )
Braking time in second t3 = V2 / β …… ( 3 )
Total distance travelled in ( km )( S ) = Distance travelled during acceleration ( Area ABC ) + Distance travelled during coasting ( Area BDEF ) + Distance travelled during retardation ( DEF )
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S = ( ½ V1t1 / 3600 ) + { ( V1 + V2 ) / 2 × ( t2 / 3600 ) } + ( ½ V2t3 / 3600 )
S = ( V1t1 / 7200 ) + { ( V1t2 / 7200 ) + V2t2 / 7200 ) } + ( V2t3 / 7200 )
S = { V1 ( t1 + t2 ) / 7200 } + { V2 ( t2 + t3 ) / 7200 }
As t1 + t2 + t3 = T
S = { V1 ( T – t3 ) / 7200 } + { V2 ( T – t1 ) / 7200 }
S = V1T / 7200 + V2T / 7200 – V1t3 / 7200 – V2t1 / 7200
Now t = V1 / α and t3 = V2 / β
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S = { T ( V1 + V2 ) / 7200 } – ( V1 × V2 ) / 7200 β – ( V1 × V2 ) / 7200 α
S = { T ( V1 + V2 ) / 7200 } – V1V2 / 7200 β – V1V2 / 7200 α
7200S = T ( V1 + V2 ) – V1V2 ( 1 / α + 1 / β )
From equation ( 2 )
t2 βC = V1 – V2
V2 = V1 – t2 βC
V2 = V1 – βC ( T – t1 – t3 )
V2 = V1 – βC ( T – V1 / α – V2 / β )
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V2 = V1 – βC T + βC V1 / α + βC V2 / β
V2 – βC V2 / β = V1 – βC T + βC V1 / α
V2 ( 1 – βC / β ) = V1 – βC ( T + V1 / α )
V2 = V1 – βC ( T + V1 / α ) / ( 1 – βC / β )
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