1.13 Speed

Motion is the change of position with respect to time. Speed quantifies this change in position, but notably without direction. It tells us exactly : how rapidly this change is taking place with respect to time.

Motion

Speed is the rate of change of distance with respect to time and is expressed as distance covered in unit time.

Evaluation of ratio of distance and time for finite time interval is called “average” speed, where as evaluation of the ratio for infinitesimally small time interval, when Δt-->0, is called instantaneous speed. In order to distinguish between average and instantaneous speed, we denote them with symbols $v_a$ and v respectively.

Determination of speed allows us to compare motions of different objects. An aircraft, for example, travels much faster than a motor car. This is an established fact. But, we simply do not know how fast the aircraft is in comparison to the motor car. We need to measure speeds of each of them to state the difference in quantitative terms.

Speed is defined in terms of distance and time, both of which are scalar quantities. It follows that speed is a scalar quantity, having only magnitude and no sense of direction. When we say that a person is pacing at a speed of 3 km/hr, then we simply mean that the person covers 3 km in 1 hour. It is not known, however, where the person is actually heading and in which direction.

Dimension of speed is $\hspace{0.33em}\mathbf{L}{\mathbf{T}}_{}^{-1}\hspace{0.33em}$ and its SI unit is meter/second (m/s).

Some values of speed

• Light : $3x{10}^8$ m/s
• Sound : 330 m/s
• Continental drift : ${10}^{-9}$ m/s

Distance .vs. time plots

Motion of an object over a period of time may vary. These variations are conveniently represented on a distance - time plot as shown in the figure.

The figure above displays distance covered in two equal time intervals. The vertical segment DB and FC parallel to the axis represents distances covered in the two equal time intervals. The distance covered in two equal time $\Delta t$ intervals may not be equal as average speeds of the object in the two equal time intervals may be different.

$$\begin{array}{c}{s}_1=v_1\Delta t=\mathrm{DB\hspace{0.5em}}\end{array}$$

$$\begin{array}{c}{s}_2=v_2\Delta t=\mathrm{FC\hspace{0.5em}}\end{array}$$

and $$\begin{array}{c}\mathrm{DB}\ne \text{FC}\end{array}$$

The distance - time plot characterizes the nature of distance. We see that the plot is always drawn in the first quadrant as distance can not be negative. Further, distance – time plot is ever increasing during the motion. It means that the plot can not decrease from any level at a given instant. When the object is at rest, the distance becomes constant and plot is a horizontal line parallel to time axis. Note that the portion of plot with constant speed does not add to the distance and the vertical segment representing distance remains constant during the motion.

Average speed

Average speed, as the name suggests, gives the overall view of the motion. It does not, however, give the details of motion. Let us take the example of the school bus. Ignoring the actual, let us consider that the average speed of the journey is 50 km/ hour. This piece of information about speed is very useful in planning the schedule, but the information is not complete as far as the motion is concerned. The school bus could have stopped at predetermined stoppages and crossings, besides traveling at different speeds for variety of reasons. Mathematically,