First slide

Motion in a plane

Question

The velocity of a particle is given by $v = v_{0} \sin ω t$ $v_{0}$ is constant and $ω = 2 π / T .$ Find the average velocity in time interval t = 0 to t = T/2.

Difficult

A

$2 v_{0}$
B

$\frac{v_{0}}{2}$
C

$\frac{v_{0}}{π}$
D

$\frac{2 v_{0}}{π}$

Solution

$v = v_{0} \sin ω t$
$\bar{v} = \frac{\int_{0}^{T / 2} v_{0} \sin (ω t) d t}{\int_{0}^{T / 2} d t}$
$= \frac{v_{0} \frac{| - \cos (ω t) |_{0}^{T / 2}}{ω}}{\frac{T}{2}}$
$= \frac{2 v_{0}}{ω T} [\{- \cos (\frac{ω T}{2})\} - {- \cos (0)}]$
$= \frac{2 v_{0}}{2 π} (- \cos π + 1)$
$= \frac{v_{0}}{π} {- (- 1) + 1}$
$(\cos π = \cos 180 ° = - 1)$
$= \frac{2}{π} v_{0}$

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