A train moving with a velocity of u m/s at t=0 is given a constant acceleration of a m/s^2 to attain some maximum velocity for some duration and then decelerates at the rate of a m/s^2 to attain the same initial velocity u m/s at t=T.Prove that the train has been running at the maximum velocity for the duration of
T[1-4(v'-u)/aT]^1/2,where v' is the average velocity of the train for the entire journey.

Question

A train moving with a velocity of u m/s at t=0 is given a constant acceleration of a m/s^2 to attain some maximum velocity for some duration and then decelerates at the rate of a m/s^2 to attain the same initial velocity u m/s at t=T.Prove that the train has been running at the maximum velocity for the duration of 
T[1-4(v'-u)/aT]^1/2,where v' is the average velocity of the train for the entire journey.

anonymous · Answer

Firstly i'd recommend drawing a diagram of velocity vs time. Let t' be the time period you are trying to find. Take each stage of the motion in turn and use the formula for distance travelled under constant acceleration to obtain an expression for the total distance travelled. Write this expression in terms of the given variables and t'. Then v' is the total distance divided by the total time. The expression for v' can then be rearranged with a little work to give the desired expression for t'.

anonymous · Answer

thanks a lot brainstorm