A train starts at a station and stops at another station. First the velocity of the train increases at a uniform rate, reaches a maximum velocity v and then starts decreasing at a constant rate. If the time spent for the entire journey is t, prove that the distance between the entire station is 1/2vt.

Dear Student,

Velocity increases or decreases with uniform rates so acceleration constant during motion. Therefore we can apply newton's equations.

So for entire journey 
S=  12at2                    ...(1)Now for ACjourneya = dvdt = v-0t'x = 12at'2 = vt'2  ....(1)and v2 = 2a(S-x)(S-x) 2vt''= v2 vt''2 = S-x    ...(2)Add eqution 1 and 2S = 12v(t'+t'') = 12vt


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