When a person moves straight and then runs in a circle around the starting point, their velocity does not remain constant due to changes in direction. Velocity is defined as displacement over time and is a vector quantity, meaning it includes both speed and direction. As the person changes direction while running in a circle, their velocity changes despite the speed potentially remaining constant. The discussion clarifies that if the person is on the right side of the circle, their velocity would be expressed as displacement over time in that direction. Overall, the key takeaway is that any change in direction alters the velocity.