When a particle moves in a circle with a uniform speed

Options

A. its velocity and acceleration re both constant
B. its velocilty is constant but the acceleration changes
C. its acceleration is constant but the velocity changes
D. its velocity and acceleration both change.

Answer

To solve the question regarding the motion of a particle moving in a circle with uniform speed, we can break down the concepts of velocity and acceleration in the context of circular motion. Here’s a step-by-step solution:

### Step 1: Understand Uniform Circular Motion
- A particle moving in a circle with uniform speed means that it travels along a circular path at a constant speed. However, it is important to note that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector changes continuously.

**Hint:** Remember that velocity is a vector quantity, which means it has both magnitude and direction.

### Step 2: Analyze Velocity
- In uniform circular motion, although the speed is constant, the direction of the velocity vector is continuously changing as the particle moves along the circular path. Therefore, the velocity of the particle is not constant because its direction is changing.

**Hint:** Think about how velocity changes when only the direction changes, even if the speed remains the same.

### Step 3: Analyze Acceleration
- The only acceleration present in uniform circular motion is the centripetal acceleration, which is directed towards the center of the circle. The magnitude of this centripetal acceleration is given by the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the speed of the particle and \( r \) is the radius of the circular path.

**Hint:** Recall that acceleration is also a vector quantity, and its direction is crucial in determining if it is constant or variable.

### Step 4: Conclusion on Velocity and Acceleration
- Since the velocity changes due to the changing direction, it is variable. Similarly, the direction of the centripetal acceleration also changes as the particle moves around the circle, making the acceleration variable as well.

**Hint:** Consider how both velocity and acceleration are affected by the changing direction in circular motion.

### Final Answer
In conclusion, when a particle moves in a circle with uniform speed, both its velocity and acceleration are variable due to the continuous change in direction.

---