Theory:

A graph is used to make information interesting and easy to understand. In the newspapers and magazines, the information is shown in the various forms of graphs such as bar graphs, pie charts and line graphs. For instance, scores of each over in a one-day cricket match are easily depicted in vertical bar graphs.
 
line graph.png
A line graph
 
Usually, in mathematics, a straight-line graph is used for solving linear equations with two variables. Likewise, to describe the motion of an object, a straight-line graph is used. Physical quantities like distance or velocity on the line graphs show dependence on another quantity, such as time.
Distance - time graph:
A distance-time graph is a form of a line graph.
A distance-time graph shows the change in the distance covered by an object with respect to time. It is a pictorial representation of the motion of objects. We choose the convenient scales to plot the axes of the graph.
Here, the quantity time is taken along the \(x\)-axis, and the distance is taken along the \(y\)-axis.
The distance-time graphs are plotted for different scenarios like objects moving with speed at a uniform rate, non-uniform rate, objects at rest, etc.
 
Fig 8.3.svg
Object moving with uniform speed
  
An object covering an equal distance in an equal interval of time travels with uniform speed. Hence, for an object with uniform speed, the distance covered by an object is directly proportional to the time taken. As a result, we get a straight line graph. The increasing distance at a uniform rate is marked by the line \(OB\).
 
The graph can be plotted for uniform velocity by changing displacement instead of distance. Changing the magnitude along the \(y\)-axis will depict the graph for an object travelling at a uniform velocity.
 
The speed of an object can also be determined using the distance-time graph.
 
Plotting slope on a graph:
  1. A small portion \(AB\) on the distance-time graph is considered.
  2. From point \(A\), a line parallel to the \(x\)-axis is drawn.
  3. From point \(B\), another line is drawn parallel to the \(y\)-axis.
  4. The intersection of these two lines at point \(C\) forms a triangle \(ABC\) in the graph.
  5. The time interval (\(t_{2}\ – t_{1}\)) is denoted by \(AC\), and the distance (\(s_{2}\ – s_{1}\)) is denoted by \(BC\).
  6. It is observed that the object covers a distance (\(s_{2}\ – s_{1}\)) in time interval (\(t_{2}\ – t_{1}\)) while moving from point \(A\) to \(B\).
The speed (\(v\)) of the object is given as,
 
v =s2s1t2t1
 
To know more about the distance-time graph, click here.