# 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.

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

So for entire journey

$S=\frac{1}{2}a{t}^{2}...\left(1\right)\phantom{\rule{0ex}{0ex}}NowforAC\hspace{0.17em}journey\phantom{\rule{0ex}{0ex}}a=\frac{dv}{dt}=\frac{v-0}{t\text{'}}\phantom{\rule{0ex}{0ex}}x=\frac{1}{2}at{\text{'}}^{2}=\frac{vt\text{'}}{2}....\left(1\right)\phantom{\rule{0ex}{0ex}}and{v}^{2}=2a(S-x)\phantom{\rule{0ex}{0ex}}(S-x)\frac{2v}{t\text{'}\text{'}}={v}^{2}\Rightarrow \frac{vt\text{'}\text{'}}{2}=S-x...\left(2\right)\phantom{\rule{0ex}{0ex}}Addeqution1and2\phantom{\rule{0ex}{0ex}}S=\frac{1}{2}v(t\text{'}+t\text{'}\text{'})=\frac{1}{2}vt$

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