A train stopping at two stations d distance apart takes time t on the journey from one station to other. Assuming that its motion is first of uniform acceleration α and then immediately of uniform retardation β, show that 1/α+1/β=t^2/2d
Train starts from rest
acceleration = α
Time T₁ for acceleration
Velocity = αT₁
Time T₂ for retardation
0 = αT₁ - βT₂
=> αT₁ = βT₂ = V
Distance in acceleration = (0 + αT₁ )/2 * T₁ = αT₁ ²/2 = V²/2α
Distance in retardation = βT₂ ²/2 = V²/2β
Total Distance =V²/2α + V²/2β = d
=>V²/α + V²/β = 2d
d = Area under graph = (1/2)(T) * V
=> V = 2d/T
=> 4d²/T²α + 4d²/T²β = 2d
=> 2d/α + 2d/β = T²
=>1/α + 1/β = T²/2d
