A wheel of radius `r` rolls without slipping with a speed `v` on a horizontal road. When it is at a point `A` on the road, a small blob of mud separates from'the wheel at- its highest point and lands at point `B` on the road:

Options

A. `AB = v sqrt((r)/(g))`
B. `AB = 2v sqrt((r)/(g))`
C. `AB = 4v sqrt((r)/(g))`
D. If `v gt sqrt(4rg)`, the blob mud will land on the wheel and not on the road

Answer

To solve the problem of finding the distance between points A and B when a blob of mud separates from a rolling wheel, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Scenario**:
- A wheel of radius \( r \) rolls without slipping on a horizontal road with a speed \( v \).
- At point A (the point of contact with the road), a blob of mud separates from the highest point of the wheel and lands at point B.

2. **Identify the Initial Conditions**:
- The wheel is rolling with a linear speed \( v \).
- The height from which the blob of mud falls is equal to the diameter of the wheel, which is \( 2r \).

3. **Determine the Time of Flight**:
- The blob of mud will fall under the influence of gravity. The vertical displacement \( s \) is given by:
\[ s = -2r \]
- The initial vertical velocity \( u_y = 0 \) (since it starts falling from rest).
- The acceleration \( a_y = -g \) (acceleration due to gravity).
- Using the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- Substituting the known values:
\[ -2r = 0 \cdot t + \frac{1}{2} (-g) t^2 \]
- This simplifies to:
\[ -2r = -\frac{1}{2} g t^2 \implies t^2 = \frac{4r}{g} \implies t = 2\sqrt{\frac{r}{g}} \]

4. **Calculate the Horizontal Distance**:
- The horizontal distance \( x \) that the blob travels while falling can be calculated using the horizontal velocity.
- The horizontal velocity \( u_x \) at the point of separation is equal to the linear speed of the wheel, which is \( v \).
- The horizontal distance \( x \) is given by:
\[ x = u_x \cdot t \]
- Substituting the values:
\[ x = v \cdot \left(2\sqrt{\frac{r}{g}}\right) = 2v\sqrt{\frac{r}{g}} \]

5. **Final Result**:
- Therefore, the distance between points A and B is:
\[ x = 4v\sqrt{\frac{r}{g}} \]