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6.2: The description of motion in the Galilean model

The variables describing motion—position, velocity, and acceleration, have direction and magnitude, and so are usefully represented as vectors. The variables we use to specify motion have precise, technical meanings. Often these meanings are different from the everyday meanings we associate with the same words. It is, of course, crucial to understand both the technical and everyday meanings of these words, and how they differ.

Position (technical meaning)

The position vector represents the position of something with respect to the origin of a particular reference frame. The position vector specifies both the distance (the length of the vector) of the object from the origin and the direction in space with respect to the origin. The position vector is usually represented with a lowercase letter \(r\).

Displacement (technical meaning)

The displacement of some object is the change in position of the object.

\[ \Delta r = r_f – r_i  \]

The differential displacement is \(dr\).

Velocity (technical meaning)

Velocity usually means the instantaneous velocity, rather than an average velocity over some time interval. When we use the word velocity without a modifier, it will mean instantaneous velocity. By definition, velocity is the time rate of change of position:

\[v = \dfrac{dr}{ dt} ,  \]

or

\[v = \lim_{ \Delta t \to 0} \dfrac{ \Delta r}{ \Delta t} . \]

When the time interval, \( \Delta \)t, is finite, we have the average velocity:

\[ v_{average} = \dfrac{ \Delta r}{ \Delta t} .  \]

Acceleration (technical meaning)

Acceleration usually means the instantaneous acceleration, rather than an average acceleration over some time interval. When we use the word acceleration without a modifier, it will mean instantaneous acceleration. By definition, acceleration is the time rate of change of velocity:

\[a = \dfrac{dv}{ dt} ,  \]

or

\[a = \lim_{ \Delta t \to 0} \dfrac{ \Delta v}{ \Delta t} \]

When the time interval, \(\Delta\)t, is finite, we have the average acceleration:

\[a_{average} = \dfrac{\Delta v}{ \Delta t} .  \]

Note that acceleration is any change in the velocity. The change can be a change in the magnitude of the velocity vector, either increasing or decreasing in length, that is, either slowing down or speeding up. The change can also be a change in direction of the velocity, with or without a change in the speed.

Everyday or Common Definitions

Distance in an everyday sense usually means how far apart two points are, and does not imply a direction, nor does it imply position. It is always a positive number. In terms of the technical motion variables, distance is the magnitude of displacement. However, the term “total distance” does not necessarily mean the magnitude of the displacement. Consider an object that moves back and forth, such as a mass on a spring. The “total distance” usually means how far the mass traveled in going back and forth, while the displacement is simply the distance between where it started and where it ended.

Speed in everyday use usually means how fast something is moving. It also does not imply direction and is always positive. In terms of the technical motion variables, speed is the magnitude (the “length”) of the velocity vector.

In everyday use, the word acceleration usually means speeding up, but does not mean slowing down. Remember, our technical definition includes both speeding up and slowing down as well as changing direction.