First slide

Average velocity

Question

A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is:

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{4}$
C

$\frac{4}{3}$
D

$\frac{3}{2}$

Solution

${<v>}_{time} = \frac{\int vdt}{\int dt} = \frac{\int_{0}^{T} at dt}{\int_{0}^{T} dt} = \frac{aT}{2}$
$< v >_{space} = \frac{\int vds}{\int ds} = \frac{\int v (\frac{ds}{dt}) dt}{\int (\frac{ds}{dt}) dt}$
$= \frac{\int_{0}^{T} v^{2} dt}{\int_{0}^{T} vdt}$
$= \frac{\int_{0}^{T} a^{2} t^{2} dt}{\int_{0}^{T} at dt} = \frac{2}{3} aT$
so, $\frac{< v >_{space}}{< v >_{time}} = \frac{2 aT / 3}{aT / 2} = \frac{4}{3}$

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