Question

A point moves in the $x-y$ plane according to $x=Asin\omega t$ and $y=A\left(1 - \text{cos} \omega t\right)$ where $A$ and $\omega $ are positive constants. The distance traversed by the point during the time $\tau$ is

• A. $\frac{A \omega ^{2}}{\tau}$
• B. $A\omega ^{2}\tau$
• C. $A\omega \tau$
• D. $\frac{A \omega }{\tau}$

Answer 1

Answer: (C)

Distance travelled by point in time t is:
$S=\displaystyle \int _{0}^{\tau} \left|\overset{ \rightarrow }{V}\right|dt$
$\because \left|\overset{ \rightarrow }{V}\right|=\sqrt{V_{x}^{2} + V_{y}^{2}}$
Where, $V_{x}=\frac{d x}{d t}=A\omega cos \omega t and$ $V_{y}=\frac{d y}{d t}=A\omega sin \omega t$
$S=\displaystyle \int _{0}^{\tau} A \omega d t=A\omega \tau$