A car speeds up in a circular path. Which of the following figures illustrates the acceleration of the car?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which figure illustrates the acceleration of a car speeding up in a circular path, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: The car is moving in a circular path and is speeding up. This means that the car has two components of acceleration: centripetal acceleration (which is directed towards the center of the circular path) and tangential acceleration (which is in the direction of the velocity and indicates an increase in speed).

2. **Identify the Components of Acceleration**:
- **Centripetal Acceleration (AC)**: This is given by the formula \( AC = \frac{V^2}{R} \), where \( V \) is the speed of the car and \( R \) is the radius of the circular path. This acceleration always points towards the center of the circle.
- **Tangential Acceleration (AT)**: This is responsible for the change in the speed of the car and is directed along the path of motion (in the direction of the velocity).

3. **Resultant Acceleration**: The total acceleration of the car is the vector sum of the centripetal and tangential accelerations. Since these two accelerations are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the resultant acceleration:
\[ A = \sqrt{AC^2 + AT^2} \]
The direction of the resultant acceleration will be at an angle to the radius of the circular path, between the centripetal and tangential accelerations.

4. **Analyze the Options**:
- **Option A**: Shows only centripetal acceleration. This is incorrect because it does not account for the tangential acceleration due to the increase in speed.
- **Option B**: Illustrates the resultant acceleration increasing in magnitude and changing direction as the car speeds up. This is consistent with the behavior of the car as it accelerates in a circular path.
- **Option C**: Incorrectly shows acceleration directed towards the center, which is not possible when the speed is increasing.
- **Option D**: Suggests that the lengths of both accelerations are equal, which contradicts the fact that the tangential acceleration must increase as the car speeds up.

5. **Conclusion**: Based on the analysis, the correct option that illustrates the acceleration of the car speeding up in a circular path is **Option B**.