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  4. A body starts from rest and travels a distance S with uniform acceleration, then moves uniformly a distance 2S and finally comes to rest after moving further 5S under uniform retardation. The ratio of the average velocity to maximum velocity is

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Q. A body starts from rest and travels a distance S with uniform acceleration, then moves uniformly a distance 2S and finally comes to rest after moving further 5S under uniform retardation. The ratio of the average velocity to maximum velocity is

AIIMSAIIMS 2018Motion in a Straight Line

Solution:

Graphically, the area of v-t curve represents displacement:
$S=\frac{1}{2}\,v_{max} \,t_{1}\quad or\quad t_{1}\,=\frac{2S}{v_{max}}$
$2S=v_{max}\,t_{2}\quad or\quad t_{2}=\frac{2S}{v_{max}};$
$5S=\frac{1}{2}\,v_{max} \,t_{3}\quad or\quad t_{3}\,=\frac{10S}{v_{max}}$
$v_{av}=\frac{Total displacement}{Total time}$
$=\frac{S+2S+5S}{\frac{2S}{v_{max}}+\frac{2S}{v_{max}}+\frac{10S}{v_{max}}}$
$\frac{v_{av}}{v_{max}}=\frac{8S}{14S}=\frac{4}{7}$

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