2.3 Time, velocity, and speed

Learning objectives

By the end of this section, you will be able to:

• Explain the relationships between instantaneous velocity, average velocity, instantaneous speed, average speed, displacement, and time.
• Calculate velocity and speed given initial position, initial time, final position, and final time.
• Derive a graph of velocity vs. time given a graph of position vs. time.
• Interpret a graph of velocity vs. time.

The information presented in this section supports the following AP® learning objectives and science practices:

• 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)
• 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1)

There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner's speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion.

Time

As discussed in Physical Quantities and Units , the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple— time    is change , or the interval over which change occurs. It is impossible to know that time has passed unless something changes.

The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events.

How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M. , so that the elapsed time would be 50 min. Elapsed time $\mathrm{\Delta }t$ is the difference between the ending time and beginning time,

where $\mathrm{\Delta }t$ is the change in time or elapsed time, $t_{\mathrm{f}}$ is the time at the end of the motion, and $t_0$ is the time at the beginning of the motion. (As usual, the delta symbol, $\mathrm{\Delta }$ , means the change in the quantity that follows it.)

Life is simpler if the beginning time $t_0$ is taken to be zero, as when we use a stopwatch. If we were using a stopwatch, it would simply read zero at the start of the lecture and 50 min at the end. If $t_0=0$ , then $\mathrm{\Delta }t=t_{\mathrm{f}}\equiv t$ .