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We have noted that distance traveled can be greater than displacement. So average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a store and return home in half an hour, and your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity.

A house and a store, with a set of arrows in between showing that the distance between them is 3 point 0 kilometers and the total distance traveled, delta x total, equals 0 kilometers.
During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero.

Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in [link] . (Note that these graphs depict a very simplified model    of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.)

Three line graphs. First line graph is of position in kilometers versus time in hours. The line increases linearly from 0 kilometers to 6 kilometers in the first 0 point 25 hours. It then decreases linearly from 6 kilometers to 0 kilometers between 0 point 25 and 0 point 5 hours. Second line graph shows velocity in kilometers per hour versus time in hours. The line is flat at 12 kilometers per hour from time 0 to time 0 point 25. It is vertical at time 0 point 25, dropping from 12 kilometers per hour to negative 12 kilometers per hour. It is flat again at negative 12 kilometers per hour from 0 point 25 hours to 0 point 5 hours. Third line graph shows speed in kilometers per hour versus time in hours. The line is flat at 12 kilometers per hour from time equals 0 to time equals 0 point 5 hours.
Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.

Making connections: take-home investigation—getting a sense of speed

If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own:

  • calculate typical car speeds in meters per second
  • estimate jogging and walking speed by timing yourself; convert the measurements into both m/s and mi/h
  • determine the speed of an ant, snail, or falling leaf

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 size 12{x rSub { size 8{f} } =x rSub { size 8{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.

distance time = 80 miles 105 minutes size 12{ { {"distance"} over {"time"} } = { {"80 miles"} over {"105 minutes"} } } {}
80 miles 105 minutes × 5280 feet 1 mile × 1 meter 3 . 28 feet × 1 minute 60 seconds = 20 m/s
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Section summary

  • Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is
    Δ t = t f t 0 , size 12{Δt=t rSub { size 8{f} } - t rSub { size 8{0} } } {}
    where t f size 12{t rSub { size 8{f} } } {} is the final time and t 0 size 12{t rSub { size 8{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 size 12{t} {} .
  • Average velocity v - size 12{ { bar {v}}} {} is defined as displacement divided by the travel time. In symbols, average velocity is
    v - = Δ x Δ t = x f x 0 t f t 0 . size 12{ { bar {v}}= { {Δx} over {Δt} } = { {x rSub { size 8{f} } - x rSub { size 8{0} } } over {t rSub { size 8{f} } - t rSub { size 8{0} } } } "." } {}
  • The SI unit for velocity is m/s.
  • Velocity is a vector and thus has a direction.
  • Instantaneous velocity v size 12{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.

Questions & Answers

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.077 m2, and the magnitude of the fluid velocity is 3.50 m/s. (a) What is the fluid speed at points in the pipe where the cross
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