A particle moving with uniform speed in a circular path maintains:

Varying velocity and varying acceleration

The correct answer is option (4) : Varying velocity and varying acceleration

When a particle moves along a circular path with a uniform speed, it is important to understand the motion characteristics in terms of both velocity and acceleration. Velocity is a vector quantity which means it has both magnitude and direction, while acceleration is the rate of change of velocity with respect to time.

First, let's analyze the velocity:

Although the speed (magnitude of the velocity vector) remains constant in uniform circular motion, the direction of the velocity vector continuously changes as the particle progresses along the circle. Since velocity includes both the magnitude and the direction, any change in either results in a change in velocity. Consequently, in uniform circular motion, the velocity of the particle is not constant but varies due to the continuous change in direction.

Next, consider the acceleration:

In circular motion, there is always an acceleration directed towards the center of the circle, known as centripetal acceleration. This acceleration is responsible for changing the direction of the velocity vector, thereby keeping the particle moving in a circle, despite the speed being constant. The formula for centripetal acceleration is:

 $a_c=\frac{v^2}{r}$

where $v$  is the speed of the particle and $r$  is the radius of the circle. This acceleration is always directed towards the center of the circle and varies with the square of the speed and inversely with the radius of the circle.

Therefore, given that both the velocity and acceleration change, the correct choice is:

 Option 4 - Varying velocity and varying acceleration

It states that in uniform circular motion, both the velocity and acceleration of the particle vary – velocity due to continuous changes in direction and acceleration due to its consistent inward (centripetal) direction towards the center of the circle.