The study of Quadrilateral speed time curve is given below.

Let

α = Acceleration in km per hour per
second

β_{C} = Coasting retardation in km per hour
per second

β = Braking retardation in km per hour per second

V_{1} = Maximum speed at the end of
acceleration ( km per hour )

V_{2} = Speed at the end of coasting period (
km per hour )

T = Total time of run ( second )

Acceleration time in second t_{1} = V_{1}
/ α …. ( 1 )

Coasting time
in second t_{2}
= V_{1}
– V_{2}
/ β_{C} ….. ( 2 )

Braking time in
second t_{3}
= V_{2}
/ 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 )

S = ( ½ V_{1}t_{1} / 3600 ) + { (
V_{1}
+ V_{2}
) / 2 × ( t_{2}
/ 3600 ) } + ( ½ V_{2}t_{3} / 3600 )

S = ( V_{1}t_{1} / 7200 ) + { (
V_{1}t_{2} / 7200 ) + V_{2}t_{2}
/ 7200 ) } + ( V_{2}t_{3}
/ 7200 )

S = { V_{1} ( t_{1} + t_{2} ) / 7200 }
+ { V_{2} ( t_{2} + t_{3} ) / 7200 }

As t_{1} + t_{2} + t_{3} = T

S = { V_{1} ( T – t_{3} ) / 7200 }
+ { V_{2 }( T – t_{1} ) / 7200 }

S = V_{1}T / 7200 + V_{2}T / 7200 – V_{1}t_{3} / 7200 – V_{2}t_{1} / 7200

Now t = V_{1 }/
α and t_{3}
= V_{2}
/ β

S = { T ( V_{1}
+ V_{2} ) / 7200 } – ( V_{1}
× V_{2} ) / 7200 β – ( V_{1} × V_{2} ) / 7200 α

S = { T ( V_{1}
+ V_{2} ) / 7200 } – V_{1}V_{2}
/ 7200 β – V_{1}V_{2} /
7200 α

**7200S = T ( V _{1}
+ V_{2} ) – V_{1}V_{2} ( 1 / α + 1 / β )**

From equation (
2 )_{ }

t_{2 }β_{C }= V_{1}
– V_{2}

V_{2} = V_{1} – t_{2}
β_{C}

V_{2} = V_{1} – β_{C}
( T – t_{1} – t_{3 })

V_{2} = V_{1} – β_{C}
( T – V_{1} / α – V_{2} / β )

V_{2} = V_{1} – β_{C}
T + β_{C} V_{1} / α + β_{C} V_{2} / β

V_{2} – β_{C} V_{2}
/ β = V_{1 }– β_{C} T +
β_{C} V_{1} / α

V_{2} ( 1 – β_{C} / β
) = V_{1} – β_{C} ( T + V_{1} / α )

**V _{2} = V_{1} – β_{C}
( T + **

**V**

_{1}/**α ) / ( 1 – β**

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