A car accelerates from rest at a constant rate α for some time after which it decelerates at a constant rate β to come to rest. If the total time elapsed is t seconds, the total distance travelled is:

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v 0 = α t 1 and 0 = v 0 - β t 2 ⇒ v 0 = β t 2 t 1 + t 2 = t v 0 1 α + 1 β = t ⇒ v 0 = α β t α + β Distance = area of v - t graph = 1 2 × t × v 0 = 1 2 × t × α β t α + β = α β t 2 2 ( α + β )

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