The physics of the motion is the same regardless of whether an object moves along the x-axis, the y-axis, or any other straight line. Consequently, it will be convenient to write equations for a “generic axis” that we will call the s-axis. The position of an object will be represented by the symbol $s$ and its velocity by $v_s$.

Consider an object in uniform motion along the s-axis with the linear position-versus-time graph shown in FIGURE 2.4. The object’s initial position is $s_{\text{i}}$ at time $t_{\text{i}}$. The term initial position, designated with subscript i, refers to the starting point of our analysis or the starting point in a problem; the object may or may not have been in motion prior to $t_{\text{i}}$. At a later time $t_{\text{f}}$, the ending point of our analysis, the object’s final position, denoted by f, is $s_{\text{f}}$.

FIGURE 2.4 The velocity is found from the slope of the position-versus-time graph.

The object’s velocity $v_s$ along the s-axis can be determined by finding the slope of the graph:

Equation 2.2 is easily rearranged to give

Equation 2.3 tells us that the object’s position increases linearly as the elapsed time $\text{Δ}t$ increases—exactly as we see in the straight-line position graph.