Instantaneous Speed

[Chapter 2 Objectives]

BHS -> Staff -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page

Consider this situation: A person running a 40-meter race. As the gun goes off, she trips and falls down. It takes her several seconds to recover, but she finally gets up and sprints toward the finish line. Just before the finish line, she falls again, recovers, and finishes the race. Total time is 40 seconds. What was her average speed?

Clearly, her average speed is 40 meters/40 seconds = 1 meter/second. But what does this mean? A speed of 1 meter/second is the speed of a person walking. The unfortunate sprinter never walked! For most of the 40 seconds, she was stopped - for a few seconds she was running much faster than 1 m/s. This example points out that average speed doesn't tell you much about what goes on during the time interval - the only thing that matters to average speed is the total distance covered and the total time taken. Clearly, average speed doesn't have much to do with the sprinter's "speedometer reading" - also called her instantaneous speed.

An object's instantaneous speed is what its speedometer would read at that instant - if it had one. As the previous example shows, average velocity and instantaneous velocity are not the same - they are related, but the relationship is not simple-minded.

If a runner has an average speed of 1 m/s, it **doesn't
mean** that her speedometer reading (instantaneous speed) was
always 1 m/s. It **does mean** that **if**
her speedometer reading were 1 m/s at all times, she would have
covered the same distance in the same time.

Determining the exact instantaneous speed of an object is very
difficult, but it is relatively straightforward to
*approximate* its instantaneous speed.

Suppose that we want to know the instantaneous speed of a dragster just as it crosses the finish line of a quarter-mile course. Suppose the car runs a quarter mile in 6.00 seconds. As a very crude approximation, we could divide 0.25 miles by the time for the car to complete the race. (6 seconds = 1/10 minute = 1/600 hour, so speed = distance/time = .25 mi/ 1/600 hr = .25(600) mi/hr = 150 mi/hr) This approximation is much too slow. The problem is that the car is moving relatively slowly at the start of the race, and speeds up as it moves down the track.

Suppose that we set up photocells to measure the time it takes the car to travel the last 100 feet of the quarter mile. During the last 100 feet of the race, the car's speed will change much less than throughout the entire race. By approximating the finishing speed as this distance divided by time, we will get a better approximation to the true instantaneous speed. If we need a better approximation, we could determine the average speed for the last 50 feet, or even 10 feet. (What distance do drag strips actually use? Does it vary from course to course?)

You can determine the instantaneous speed of an object by determining the average speed over a short distance (and short time), and using this approximation as the instantaneous speed.

Note that if a physicist mentions "speed",
she is probably referring to *instantaneous speed* instead of
average speed

[Chapter 2 Objectives] BHS -> Staff -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page

last update January 19, 2009 by JL Stanbrough