A bug is crawling with constant speed v, along the spoke of a bicycle wheel, of radius a while the bicycle moves down the road with constant speed v. Find the accelerations of the bug as observed by a man standing beside the road, along the perpendicular to the spoke of the wheel.

Answer

To solve the problem, we need to analyze the motion of the bug as it crawls along the spoke of the bicycle wheel while the bicycle moves down the road. We will determine the acceleration of the bug as observed by a man standing beside the road.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The bug is crawling along the spoke of the wheel with a constant speed \( v \).
- The bicycle is moving down the road with a constant speed \( v \).
- The radius of the bicycle wheel is \( a \).

2. **Identifying the Components of Motion**:
- The bug's motion can be broken down into two components:
- Radial motion (along the spoke of the wheel).
- Tangential motion (due to the rotation of the wheel).

3. **Calculating Radial Acceleration**:
- The radial acceleration \( a_r \) of the bug can be calculated using the formula for radial acceleration:
\[ a_r = \frac{v^2}{a} \]
- Here, \( v \) is the speed of the bug along the spoke, and \( a \) is the radius of the wheel.

4. **Calculating Tangential Acceleration**:
- The bug also experiences tangential acceleration due to the movement of the bicycle. Since the bicycle is moving at a constant speed \( v \), the tangential acceleration \( a_t \) can be expressed as:
\[ a_t = 2 \frac{v \cdot v}{a} = \frac{2v^2}{a} \]
- This accounts for the effect of the bicycle's speed on the observed motion of the bug.

5. **Total Acceleration**:
- The total acceleration \( a \) of the bug as observed by the man standing beside the road is the vector sum of the radial and tangential accelerations. However, since we are interested in the component of acceleration perpendicular to the spoke, we can focus on the tangential component:
\[ a_{\perpendicular} = a_t = \frac{2v^2}{a} \]

### Final Answer:
The acceleration of the bug as observed by a man standing beside the road, along the perpendicular to the spoke of the wheel, is given by:
\[ a_{\perpendicular} = \frac{2v^2}{a} \]