1.3.4 Time Interval

It’s also useful to consider a change in time. For example, the clock readings of two frames of a video might be $t_{1}$ and $t_{2}$ . The specific values are arbitrary because they are timed relative to an arbitrary instant that you chose to call $t = 0$ . But the time interval $Δ t = t_{2} - t_{1}$ is not arbitrary. It represents the elapsed time for the object to move from one position to the next.

The time interval $Δ t = t_{b} - t_{a}$ measures the elapsed time as an object moves from position $\vec{r}_{a}$ at time $t_{a}$ to position $\vec{r}_{b}$ at time $t_{b}$ . The value of $Δ t$ is independent of the specific clock used to measure the times.

A photo of an analogue stopwatch. — A stopwatch is used to measure a time interval.

To summarize the main idea of this section, we have added coordinate systems and clocks to our motion diagrams in order to measure when each frame was exposed and where the object was located at that time. Different observers of the motion may choose different coordinate systems and different clocks. However, all observers find the same values for the displacements $Δ \vec{r}$ and the time intervals $Δ t$ because these are independent of the specific coordinate system used to measure them.