Position-Time Graph

Position-time graphs are the most basic form of graphs in kinematics, which allow us to describe the motion of objects. In these graphs, the vertical axis represents the position of the object while the horizontal axis represents the time elapsed: the dependent variable, position, depends on the independent variable, time. In this way, the graph tells us where the particle can be found after some amount of time. Graphs such as these help us visualize the trajectory of objects. An amazing amount can be learned by studying a position-time graph for an object, as long as we know how to properly analyze them.

We know that the slope of a function is its derivative, and that the derivative of the displacement function is the velocity function, and thus the slope of a position-time graph gives us the velocity of the object:

$$(\text{Slope of a position time graph})=v=\dfrac{s_2-s_1}{t_2-t_1}.$$

Find the velocity of the particle over the interval $(1,2)$.

We have seen that the slope of the position-time graph gives us the velocity over the time period. Thus, $$v=\dfrac{s_2-s_1}{t_2-t_1}=\dfrac{10-7.5}{2-1}=2.5\text{ (m/s)}.$$

Note: We can see that the displacement of this object is given by the function $s(t)=2.5t+5$. Thus, we can find the velocity of this function which is nothing but its derivative: $$v(t)=\dfrac{d}{dt}(2.5t+5)=2.5\text{ (m/s)}.$$