Speed and Velocity

Cut to the Chase

Speed

What's the difference between two identical objects traveling at different speeds? Nearly everyone knows that the one moving faster (the one with the greater speed) will go farther than the one moving slower in the same amount of time. Either that or they'll tell you that the one moving faster will get where it's going before the slower one. Whatever speed is, it involves both distance and time. "Faster" means either "farther" (greater distance) or "sooner" (less time). Doubling one's speed would mean doubling one's distance traveled in a given amount of time. Doubling one's speed would also mean halving the time required to travel a given distance. If you know a little about mathematics, these statements are meaningful and useful. (The symbol v is used for speed because of the association between speed and velocity, which will be discussed shortly).

Speed is directly proportional to distance when time is constant: v ∝ s (t constant)

Speed is inversely proportional to time when distance is constant: v ∝ 1/t (s constant)

Combining these two rules together gives the definition of speed in symbolic form.

Don't like symbols? Well then, here's another way to define speed.

Speed is the rate of change of distance with time.

In order to calculate the speed of an object we must know how far it's gone and how long it took to get there. "Farther" and "sooner" correspond to "faster". Let's say you drove a car from New York to Boston. The distance by road is roughly 300 km (200 miles). If the trip takes four hours, what was your speed? Applying the formula above gives …

This is the answer the equation gives us, but how right is it? Was 75 kph the speed of the car? Yes, of course it was … Well, maybe, I guess … No, it couldn't have been the speed. Unless you live in a world where cars have some kind of exceptional cruise control and traffic flows in some ideal manner your speed during this hypothetical journey must certainly vary. Thus, the number calculated above is not the speed of the car, it's the average speed for the entire journey. In order to emphasize this point, the equation is sometimes modified as follows …

The line over the v indicates an average or a mean and the delta symbols indicate a difference or a change. This is the quantity we calculated for our hypothetical trip.

In contrast, a car's speedometer shows its instantaneous speed, that is, the speed determined over a very small interval of time -- an instant. Ideally this interval should be as close to zero as possible, but in reality we are limited by the sensitivity of our measuring devices. Mentally, however, it is possible imagine calculating average speed over ever smaller time intervals until we have effectively calculated instantaneous speed. This idea is written symbolically as …

or, in the language of calculus speed is the first derivative of distance with respect to time.

On a distance-time graph, speed corresponds to slope and thus the instantaneous speed of an object with non-constant speed can be found from the slope of a line tangent to its curve. We will deal with this in a later section of this chapter.

velocity

In order for you or me to calculate the speed of an object we must know how far it goes and how long it takes to get there. Astute observers should then ask a following question …

What do you mean by "how far"? Didn't we learn in the previous section that there are two quantities used to answer the question "how far"?

My, but you are wise. Yes indeed, there are two ways to answer that question. When you ask "how far" do you mean distance or displacement? There's a difference between the two quantities and thus a difference between the two answers.

Speed is the rate of change of distance with time. Velocity is the rate of change of displacement with time.

Which means that for the calculus people …

Speed is the first derivative of distance with respect to time. Velocity is the first derivative of displacement with respect to time.

Velocity and speed mean pretty much the same thing to the average English speaking person, but physics is more precise in its language than is everyday speech.

The situation is not entirely hopeless, however. All the types of speed discussed above also have their counterparts in velocity. Just replace the symbol for distance with the symbol for displacement et voila, you've got velocity.

	Equations for Speed
average
instantaneous

	Equations for Velocity
average
instantaneous

Speed and velocity are related in much the same way that distance and displacement are related. Displacement is measured along the shortest path between two points and thus its magnitude is always less than or equal to the distance. The magnitude of the displacement approaches the distance as distance approaches zero. That is, distance and displacement are effectively the same (have the same magnitude) when the interval examined is "small". Since speed is based on distance and velocity is based on displacement, these two quantities are effectively the same (have the same magnitude) when the time interval examined is "small" or, in the language of calculus the magnitude of an object's average velocity approaches its average speed as the time interval approaches zero.

Thus, the instantaneous speed of an object is the magnitude of its instantaneous velocity.

v = |v|

units

Speed and velocity are both measured using the same units. Given that the SI unit of both distance and displacement is the meter and that the SI unit of time is the second, it should be intuitively obvious that the unit of both speed and velocity would be a ratio of two units. The SI unit of speed and velocity is the meter per second [m/s].

This unit is only rarely used outside scientific and academic circles. Most people on this planet measure speeds in kilometer per hour (km/h or sometimes kph). The United States is an exception in that we use the comparatively archaic mile per hour (mi/h or mph). Let's determine the conversion factors so that we can relate speeds measured in m/s with the more familiar, everyday units.

The decimal values are accurate to four significant digits, but the fractional values should only be considered rules of thumb (1 mph is really more like 4/10 m/s).

The ratio of any unit of distance to any unit of time is a unit of speed.

Summary

• Speed is the rate of change of distance with time.
  • As a scalar it has magnitude only.
  • Average speed is speed measured over a non-zero time interval.
  • Instantaneous speed is the limit of average speed as the time interval approaches zero.
  • Instantaneous speed is also the first derivative of distance with respect to time.
  • The symbol for speed is v (italic).
  • The symbol for average speed is v_ave or v (overline)
• Velocity is the rate of change of displacement with time.
  • As a vector it must be stated with both magnitude and direction.
  • Average velocity is velocity measured over a non-zero time interval.
  • Instantaneous velocity is the limit of average velocity as the time interval approaches zero.
  • Instantaneous velocity is also the first derivative of displacement with respect to time.
  • The symbol for velocity is v (boldface)
  • The symbol for average velocity is v_ave or v (overline)

    Speed and velocity are defined by the following equations …

    	Equations for Speed
    average
    instantaneous

    	Equations for Velocity
    average
    instantaneous

  • The SI unit of speed and velocity is the meter per second [m/s].