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.

Figure 1: Distance is given by the vertical segment parallel to the axis representing distance.

Distance time plot

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 Δ tΔ t intervals may not be equal as average speeds of the object in the two equal time intervals may be different.

s 1 = v 1 Δ t = DB  s 1 = v 1 Δ t = DB 

s 2 = v 2 Δ t = FC  s 2 = v 2 Δ t = FC 

and DB ≠ FC DB ≠ FC

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.

Figure 2: Static condition is represented by a horizontal section on distance - time plot.

Distance time plot

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,

Mathematically,

On a distance time plot, average speed is equal to the slope of the the straight line, joining the end points of the motion, makes with the time axis. Note that average speed is equal to the slope of the chord (AB) and not that of the tangent to the curve.

Figure 3: Distance is equal to the tangent of the angle that the chord of motion makes with time axis.

Distance time plot

v a = tan θ = Δ s Δ t = BC AC v a = tan θ = Δ s Δ t = BC AC

Instantaneous speed (v)

Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total distance and time interval, but with one qualification that time interval is extremely (infinitesimally) small. This qualification of average speed has important bearing on the value and meaning of speed. The instantaneous speed is the speed at a particular instant of time and may have entirely different value than that of average speed. Mathematically,

Where Δs Δs is the distance traveled in time Δt Δt .

As Δt Δt tends to zero, the ratio defining speed becomes finite and equals to the first derivative of the distance. The speed at the moment ‘t’ is called the instantaneous speed at time ‘t’.

On the distance - time plot, the speed is equal to the slope of the tangent to the curve at the time instant ‘t’. Let A and B points on the plot corresponds to the time t t and t + Δt t + Δt during the motion. As Δt Δt approaches zero, the chord AB becomes the tangent AC at A. The slope of the tangent equals ds/dt, which is equal to the instantaneous speed at 't'.

Figure 4: Instantaneous speed is equal to the slope of the tangent at given instant.

Speed time plot

v = tan θ = DC AC = ds dt v = tan θ = DC AC = ds dt

Speed - time plot

The distance covered in the small time period dt is given by :

ds = v dt ds = v dt

Integrating on both sides between time intervals t 1 t 1 and t 2 t 2 ,

The right hand side of the integral represents an area on a plot drawn between two variables, speed (v) and time (t). The area is bounded by (i) v-t curve (ii) two time ordinates t 1 t 1 and t 2 t 2 and (iii) time (t) axis as shown by the shaded region on the plot.

Figure 5: Area under v-t plot gives the distance covered by the object in a given time interval.

Area under v-t plot Area under v-t plot Subfigure 5.1 Subfigure 5.2

Alternatively, we can consider the integral as the sum of areas of small strip of rectangular regions (vxdt), each of which represents the distance covered (ds) in the small time interval (dt). As such, the area under speed - time plot gives the total distance covered in a given time interval.

Position - time plot

The position – time plot is similar to distance – time plot in one dimensional motion. In this case, we can assess distance and speed from a position - time plot. Particularly, if the motion is unidirectional i.e. without any reversal of direction, then we can substitute distance by position variable. The example here illustrates this aspect of one-dimensional motion.

Example 2

Problem The position – time plot of a particle’s motion is shown below.

Figure 6

Position - time plot

Determine :

• average speed in the first 10 seconds
• Instantaneous speed at 5 seconds
• Maximum speed speed
• The time instant(s), when average speed measured from the beginning of motion equals instantaneous speed

Solution

1. Average speed in the first 10 seconds

The particle covers a distance of 110 m. Thus, average speed in the first 10 seconds is :

v a = 110 10 = 11 m/s v a = 110 10 = 11 m/s

2. Instantaneous speed at 5 second

We draw the tangent at t = 5 seconds as shown in the figure. The tangent coincides the curve between t = 5 s to 7 s. Now,

Figure 7

Position - time plot

v = slope of the tangent = 30 2 = 15 m/s v = slope of the tangent = 30 2 = 15 m/s

3. Maximum speed

We observe that the slope of the tangent to the curve first increase to become constant between A and B. The slope of the tangent after point B decreases to become almost flat at the end of the motion. It means the maximum speed corresponds to the constant slope between A and B, which is is 15 m/s

4. Time instant(s) when average speed equals instantaneous speed

In the period of constant slope, speed is constant and hence average speed equals instantaneous speed in this period. But this period does not start from the beginning of motion. In order to find the time instant as required from the origin of the plot, we draw a straight line from the origin, which is tangent to the curve. The only such tangent is shown in the figure below. Now, this straight line is the chord between the point and origin and tangent to the curve at the point as well. Thus, average speed equals instantaneous speeds at t = 9 s.

Figure 8

Position - time plot