The displacement of the point of the wheel initially in contact with the ground, when the wheel roles forward half a revolution will be (radius of the wheel is R)

The correct option is R√(π²+4).

To find the displacement of the point on the wheel initially in contact with the ground when the wheel rolls forward half a revolution, we can analyze its horizontal and vertical motion separately.

Let the radius of the wheel be $R$. Initially, the point is at the bottom-most position, in contact with the ground.

1. Horizontal Displacement ($x$):
When the wheel completes half a revolution, the distance moved forward by the center of the wheel (and thus the horizontal distance traveled by the point) is half of the wheel's circumference:
$x=\frac{2\pi R}{2}=\pi R$

2. Vertical Displacement ($y$):
Initially, the point is at the bottom (ground level, $y=0$). After half a revolution, the point rotates to the topmost position of the wheel. The vertical distance between the lowest point and the highest point of the wheel is equal to its diameter:
$y=2R$

3. Net Displacement ($s$):
The net displacement of the point is the straight-line distance from its initial position to its final position, which can be calculated using the Pythagorean theorem:
$s=\sqrt{x^2+y^2}$

Substituting the values of $x$ and $y$:
$s=\sqrt{{\left(\pi R\right)}^2+{\left(2R\right)}^2}$

Simplifying the expression:
$s=\sqrt{{\pi }^2{R}^2+4R^2}$

$s=R\sqrt{{\pi }^2+4}$

Therefore, the displacement of the point is $R\sqrt{{\pi }^2+4}$.