A racecar, traveling at constant speed, makes one lap around a circular track of radius r in a time t .
Which one of the following statements concerning this car is true ?

Options

A. The displacement of the car does not change with time.
B. The instantaneous velocity of the car is constant.
C. The average speed of the car is same over any time interval.
D. The average velocity of the car is the same over any time interval.

Answer

To solve the problem, we need to analyze the statements regarding a racecar traveling at constant speed around a circular track. Let's break down the statements one by one.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The racecar is moving in a circular path with a constant speed. The radius of the track is \( r \), and the time taken for one complete lap is \( t \).

2. **Analyzing Statement 1**:
- **Statement**: "The displacement of the car does not change with time."
- **Analysis**: Displacement is a vector quantity that depends on the initial and final positions of the car. Since the car is moving in a circle, its position changes continuously, and thus the displacement changes as well.
- **Conclusion**: This statement is **false**.

3. **Analyzing Statement 2**:
- **Statement**: "The instantaneous velocity of the car is constant."
- **Analysis**: Instantaneous velocity is also a vector quantity. While the speed (magnitude of velocity) is constant, the direction of the velocity changes continuously as the car moves around the circular track. Therefore, the instantaneous velocity is not constant.
- **Conclusion**: This statement is **false**.

4. **Analyzing Statement 3**:
- **Statement**: "The average speed of the car is the same over any time interval."
- **Analysis**: Average speed is defined as the total distance traveled divided by the total time taken. Since the car travels at a constant speed around the circular track, the average speed will remain the same regardless of the time interval chosen.
- **Conclusion**: This statement is **true**.

5. **Analyzing Statement 4**:
- **Statement**: "The average velocity of the car is the same over any time interval."
- **Analysis**: Average velocity is defined as the total displacement divided by the total time taken. Since the displacement changes with each time interval (the car returns to its starting point after one full lap, but not before), the average velocity will not be the same over different time intervals.
- **Conclusion**: This statement is **false**.

### Final Answer:
The correct statement concerning the car is: **The average speed of the car is the same over any time interval.** (Statement 3)