Question

The velocity of a train increases at a constant rate of A from velocity 0 to v and then remains constant for some time and then finally decreases
to 0 at a constant rate of B. If the total distance covered by the train is x then show that the total time taken is t = (x/v) + (v/2)[(1/A)+(1/B)]

Answer 1

hmm

Answer 2

Time taken for train to accelerate = v/A .............(1)
Time taken for train to decelerate = v/B ............(2)
Distance travelled during acceleration = average speed * time = (v/2) * (v/A) =
\[\frac{v ^{2}}{2A}\ ..........(3)\]
Distance travelled during deceleration = average speed * time = (v/2) * (v/B) =
\[\frac{v ^{2}}{2B}\ ..........(4)\]
Distance travelled at constant speed v is given by
\[x-(\frac{v ^{2}}{2A}+\frac{v ^{2}}{2B})=x-\frac{v ^{2}}{2}(\frac{1}{A}+\frac{1}{B})\ .....(5)\]
The time travelled at constant speed v is found by dividing the distance given by equation (5) by the speed v, giving the following result
\[\frac{x}{v}-\frac{v}{2}(\frac{1}{A}+\frac{1}{B})\ ...............(6)\]
The total time taken is found by adding the times in (1) and (2) to the time in (6) as follows:
\[Total\ time=\frac{x}{v}-\frac{v}{2}(\frac{1}{A}+\frac{1}{B})+\frac{v}{A}+\frac{v}{B}=\frac{x}{2}+\frac{v}{2}(\frac{1}{A}+\frac{1}{B})\]