What is the displacement of the point of a wheel initially in contact with the ground when the wheel rolls forward half a revolution? Take the radius of the wheel as `R` and the `x`-axis as the forward direction?

Explanation

From figure, during half revolution of the wheel, the point `A` covers `AC=piR` in horizontal direction, and `BC=2R` in vertical direction
`x=piR` and `y=2R`
`AB=sqrt(x^(2)+y^(2))=sqrt((piR)^(2)+(2R)^(2))=Rsqrt(pi^(2)+4)`
and `theta=tan^(-1)(y/x)=tan^(-1)((2R)/(piR))`
`=tan^(-1)((2)/(pi))`with `x`axis.