The speed of a train increases at a constant rate from zero to \( v \) and then remains constant...

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The speed of a train increases at a constant rate from zero to \( v \) and then remains constant for an interval and finally decreases to zero at a constant rate \( \beta \). If \( l \) be the total distance described, prove that the total time taken is :
\[
\frac{l}{v}+\frac{v}{2}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)
\]
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