Question

A particle moves in the $x-y$ plane. Its motion is given by equations $x=\sin 2t$ and $y=(1-\cos 2t)$. The distance travelled by the particle during time $t=2s$ is

• A. $1m$
• B. $6m$
• C. $4m$
• D. $8m$

Answer 1

The correct option is C $4m$

Distance travelled,

$={\int }_0^2\sqrt{{\left(\frac{d}{dt}\left(x\right)\right)}^2+{\left(\frac{d}{dt}\left(x\right)\right)}^2}dt$

$={\int }_0^2\sqrt{{\left(\frac{d}{dt}\left(sin2t\right)\right)}^2+{\left(\frac{d}{dt}(1-cos2t)\right)}^2}dt$

$={\int }_0^2\sqrt{4{\cos }^{2}2t+4{\sin }^{2}2t}dt$

$={\int }_0^2\sqrt{4}dt$

$={\int }_0^{2}2dt$

$=2[t{]}_0^2$

$=2[2-0]$

$=4m$