Distance-Time Table and Distance-Time Graph

Mr.X is traveling from terminus A to terminus B in a bus and records his observation

 

Observations

Distance in km	0	10	20	30	40	50	60
Time	10.00am	10.15am	10.30am	10.45am	11.00am	11.15am	11.30am

The above table tells us that the bus is covering equal distances in equal intervals of time i.e., the bus is moving with uniform speed. If the bus continues to move with uniform speed then we can calculate the distance covered by the bus at any time.

Consider an object moving with uniform speed v from its initial position x_i to final position x_f in time t.



The equation (1) gives the relation between distance, time and average speed. This relation can be used for making distance-time tables and also to determine the position of any moving object at any given time.
But it is a long and tedious process especially when we have to determine the position after a long time or when we have to compare the motion of two objects. In such situations we can make use of graphs like distance-time graph.
A distance-time graph is a line graph showing the variation of distance with time.
In a distance-time graph, time is taken along x-axis and distance along y-axis.
Let us now plot a distance-time graph for the above example.

Distance-time graph for uniform motion

• Take time along x-axis and distance along y-axis.

• Analyze the given data and make a proper choice of scale for time and distance.

• Plot the points.

• Consider any two points (A, B) on the straight line graph.

• Draw perpendiculars from A and B to x and y axes.

• Join A to C to get a right angled triangle ACB.

Calculation:

• Write the title and scale chosen for the graph.

• Consider another two points P and Q on the graph and construct a right-angled triangle PRQ.

The distance-time graph is a straight line showing that the motion is uniform. Thus from the distance-time graph (S-t) we calculate the speed.

The ratio BC/AC is the slope of the graph. Hence, the slope of the distance-time graph gives the speed of the moving object.
In the second case also we get the speed as 0.66 km/min showing that the speed is uniform.
Now, let us see the nature of distance-time graph for a non-uniform motion. The following table gives the distance covered by a bus after every 15 minutes.

Distance covered in km	0	5	15	20	25	30	35
Time in minutes	0	15	30	45	60	75	90

From the above table we can conclude that the motion is non-uniform i.e., it covers unequal distances in equal intervals of time. The graph for a non-uniform motion will be as follows:

Distance-time graph for a non-uniform motion

• Take time along x-axis and distance along y-axis.

• Analyze the given data and make a proper choice of scale for time and distance.

• Plot the points.

• Consider any two points (A, B) on the graph.

• Draw perpendicular from A to B to x and y axes.

• Join A to C to get a right angled ACB.

Calculations:

• Write the title and scale chosen for the graph.

• Consider another two points P and Q on the graph and construct a right angled triangle PRQ.

We can infer that speed is not uniform.
Let us now see the nature of S-t graph for non-uniform motion.

Fig (a) represents the S-t graph when the speed of a moving object increases and Fig (b) represents the S-t graph when the speed of a moving object decreases.
From the nature of S-t graph we can conclude whether the object is moving with uniform speed or variable speed.

Uses of graphical representation

• Graphical representation is more informative than tables as it gives a visual representation of two quantities (e.g., speed vs. time)

At a glance a graph gives more information than a table. Both the graphs shown here represent the increasing speed.
Fig (1) gives us an idea of nature of variation of speed i.e., increase is greater in the beginning up to line t₁ and relatively lower after t₂. Similarly, fig (2) gives an idea that the increase in the speed becomes greater after t₁. Similar explanation holds good for the decreasing speed also.

• Graphs can be easily read at a glance.

• Plotting graphs takes less time and are more convenient to draw.

• By using graphs, position of any moving object at any intermediate interval can be easily determined.

• Motion of two moving objects can be easily compared.

• Graphs tell us about the nature of motion.