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There are two possibilities, when the distance time graph is horizontal.

1. The distance is always constant means the position of the object is always fixed and constant. The object is Stationary. Speed = 0. Its distance from origin is always S0.

2. The position of the object and hence the displacement changes in such a way that the distance remains constant. This is the case of uniform or non-uniform circular motion. An example, a stone tied to a string is rotated in a horizontal or vertical circle. Its distance from origin always remains constant at Radius R. Velocity and displacement change.

Horizontal line or Y axis represents Dependent quantity means distance, velocity , etc. whereas Vertical line or X axis represents independent quantity means time. as time does not wait for anyone it is independent quantity and is written on vertical line. but if there will be no time then what distance we would travel that's why it is written on y axis.