# 1.3 Time, velocity, and speed  (Page 4/12)

 Page 4 / 12

A commuter train travels from Baltimore to Washington, DC, and back in 1 hour and 45 minutes. The distance between the two stations is approximately 40 miles. What is (a) the average velocity of the train, and (b) the average speed of the train in m/s?

(a) The average velocity of the train is zero because ${x}_{f}={x}_{0}$ ; the train ends up at the same place it starts.

(b) The average speed of the train is calculated below. Note that the train travels 40 miles one way and 40 miles back, for a total distance of 80 miles.

$\frac{\text{distance}}{\text{time}}=\frac{\text{80 miles}}{\text{105 minutes}}$
$\frac{\text{80 miles}}{\text{105 minutes}}×\frac{\text{5280 feet}}{\text{1 mile}}×\frac{\text{1 meter}}{3\text{.}\text{28 feet}}×\frac{\text{1 minute}}{\text{60 seconds}}=\text{20 m/s}$

## Section summary

• Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is
$\Delta t={t}_{f}-{t}_{0},$
where ${t}_{f}$ is the final time and ${t}_{0}$ is the initial time. The initial time is often taken to be zero, as if measured with a stopwatch; the elapsed time is then just $t$ .
• Average velocity $\stackrel{-}{v}$ is defined as displacement divided by the travel time. In symbols, average velocity is
$\stackrel{-}{v}=\frac{\Delta x}{\Delta t}=\frac{{x}_{\text{f}}-{x}_{0}}{{t}_{\text{f}}-{t}_{0}}\text{.}$
• The SI unit for velocity is m/s.
• Velocity is a vector and thus has a direction.
• Instantaneous velocity $v$ is the velocity at a specific instant or the average velocity for an infinitesimal interval.
• Instantaneous speed is the magnitude of the instantaneous velocity.
• Instantaneous speed is a scalar quantity, as it has no direction specified.
• Average speed is the total distance traveled divided by the elapsed time. (Average speed is not the magnitude of the average velocity.) Speed is a scalar quantity; it has no direction associated with it.

## Conceptual questions

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.

There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.

Does a car's odometer measure position or displacement? Does its speedometer measure speed or velocity?

If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity? Under what circumstances are these two quantities the same?

How are instantaneous velocity and instantaneous speed related to one another? How do they differ?

## Problems&Exercises

(a) Calculate Earth's average speed relative to the Sun. (b) What is its average velocity over a period of one year?

(a) $3\text{.}\text{0}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(b) 0 m/s

A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation. (a) Calculate the average speed of the blade tip in the helicopter's frame of reference. (b) What is its average velocity over one revolution?

The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate how long will it take them to drift 500 km farther apart than they are at present?

$2×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{years}$

Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion remains constant?

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