Speed and velocity

Question: What is the difference between speed and velocity?

In some physics textbooks, speed is defined as the rate of change of distance with time, whereas velocity is the rate of change of displacement with time. In addition, speed is a scalar quantity and velocity is a vector quantity. Simply put, the difference between these two physical quantities is that speed does not have a direction, whereas velocity has a direction. However, the difference is also related to the concepts of distance and displacement.

To be more precise, a theoretical definition of speed is the rate of change of distance traveled by an object with respect to time in an inertial frame of reference. Similarly, velocity is the rate of change of displacement traveled by an object with respect to time in an inertial frame of reference. Essentially, the speed and velocity of the object are dependent on an observer’s reference frame. Moreover, an operational definition of speed is “what the speedometer measure.” In other words, the measured speed and velocity of the object are dependent on the type of speedometer used.

How would Feynman answer?

Feynman would provide a definition of speed, a definition of velocity, and explain the difference between speed and velocity as shown below.

1. A definition of speed:

In Feynman’s own words, “[m]any physicists think that measurement is the only definition of anything. Obviously, then, we should use the instrument that measures the speed -- the speedometer (Feynman et al., 1963, section 8–2 Speed).” In short, the measured speed of an object is dependent on the speedometer used as well as the experimental operations. This is related to operationalism (a kind of philosophy) which means that “the concept is synonymous with the corresponding set of operations’ (Bridgman 1927, p. 5).” Thus, a theoretical concept may be considered meaningless if it cannot be measured. Based on the same philosophy, some physicists prefer to define weight as “what the weighing scale measure” instead of “gravitational force on an object.”

In addition, Feynman elaborates that “we can define the speed in this way: We ask, how far do we go in a very short time? We divide the distance by the time, and that gives the speed. But the time should be made as short as possible, the shorter the better, because some change could take place during that time (Feynman et al., 1963, section 8–2 Speed).” That is, the speed of an object is a ratio of distance moved to the time interval measured. Importantly, the speed of the object is dependent on the measurement procedure such as how the time interval is measured. For example, an experimenter may measure the total distance traveled by the object and the time elapsed in one hour or in one second. Interestingly, Feynman also distinguishes the meaning of 88 feet per second and 60 miles per hour.

2. A definition of velocity:

Feynman also provides a mathematical definition of velocity: “[l]et us try to define velocity a little better. Suppose that in a short time, ϵ, the car or other body goes a short distance x; then the velocity, v, is defined as v = x/ϵ, an approximation that becomes better and better as the ϵ is taken smaller and smaller (Feynman et al., 1963, section 8–2 Speed). This definition of velocity is based on the ratio of an infinitesimal distance to the corresponding infinitesimal time. Theoretically speaking, we imagine what happens to that ratio as the time we use is shorter and shorter. In other words, we take a limit of the distance traveled divided by the time elapsed, as the time taken is assumed to be shorter and shorter, ad infinitum. This idea was independently invented by Newton and Leibnitz and it is now known as calculus.

Alternatively, Feynman mentions that “we have another law that the velocity is equal to the integral of the acceleration. This is just the opposite of a = dv/dt; we have already seen that distance is the integral of the velocity, so distance can be found by twice integrating the acceleration (Feynman et al., 1963, section 8–5 Acceleration).” That is, the velocity of an object can be determined not only by differentiation, but it can be calculated by using integration or summing the total area under a curve. Moreover, the velocity of the object can be mathematically represented as a two-dimensional quantity or three-dimensional quantity. For instance, we can represent the velocity as v = ds/dt = √(v_x² + v_y²).

3. The Difference between speed and velocity:

Feynman clarifies that “[o]rdinarily we think of speed and velocity as being the same, and in ordinary language they are the same. But in physics, we have taken advantage of the fact that there are two words and have chosen to use them to distinguish two ideas. We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction (Feynman et al., 1963, section 9–2 Speed and velocity).” In short, velocity is speed in a specified direction. Thus, we can also define velocity by describing how the x-, y-, and z-coordinates of an object change with time, as well as write v_x = Δx/Δt, v_y = Δy/Δt and v_z = Δz/Δt. (Mathematicians may disagree with physicists’ interpretation of notations such as v = dx/dt or v = Δx/Δt.)

On the other hand, there are problems in defining speed as well as determining speed accurately. For example, Feynman mentions that a speedometer may be spoilt, however, the speedometer has inherent uncertainty depending on the technologies used. In general, there are different kinds of speedometer such as an electronic speedometer, Doppler radar, and Global Positioning System (GPS) device. The measurement uncertainty of a car’s electronic speedometer is dependent on the interaction between a precision watch mechanism and a mechanical pulsator driven by the car’s wheel. The uncertainty of a Doppler traffic radar is dependent on a car’s direction in moving and the wavelength of the radar waves generated. The positional accuracy of a GPS device is dependent on the satellite signal quality and the position averaging software used by GPS to reduce errors.

References:

1. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.