Graphical Representation of Motion
Graphs is a pictorial representation of the relation between two sets of data of which one set is of dependent variables (like distance, speed, velocity or another quantity) and the other set is of independent variables (like time). For example, in the telecast of a cricket match, vertical bar graphs show the run rate of a team in each over.
To describe the motion of an object, we can use line graphs. In this case, line graphs show dependence of one physical quantity, such as distance or velocity, on another quantity, such as time.
Distance – Time Graph
The change in the position of an object with time can be represented on the distance-time graph.
In this graph,
On x-axis --- Time. On y-axis --- Distance.
Distance-time graph for
1) Uniform motion is --- a straight line.
2) Non-uniform motion is --- a curve.
3) An object at rest is --- a straight line parallel to time-axis.
Slope of the distance-time graph gives --- speed. The speed, v of the object,
can be represented as
v = (S_{2} - S_{1})/ (t_{2} - t_{1})
Calculation of Velocity using distance-time graph:
To calculate the velocity, let take two points A and B on the slope OB. Draw one line parallel to y-axis and another parallel to x-axis from B.
Again draw a line parallel to y-axis and another parallel to x-axis from point A. Let, line parallel to x-axis from point B cut at a point, S_{2} at y-axis. Line parallel to x-axis from point A cut at point, S_{1} at y-axis. Let, line parallel to y-axis from point B cut at t_{2} at x-axis. Line parallel to y-axis from point A cut at t_{1} at x-axis.
Now, BC= Distance = S_{2} – S_{1} and AC = time = t_{2} – t_{1}
We know that slope of the graph is given by the ratio of change in y-axis and change in X-axis.
or, slope = (Change in y-axis)/ (Change in x-axis)`
Thus, slope, AB = BC/ AC
or, v = (s_{2} − s_{1})/ (t_{2}−t_{1})
Where, v = velocity, (s_{2 }− s_{1}) = interval of distance and (t_{2} − t_{1}) = time interval
Thus, velocity = Distance/ Time
Velocity-Time Graph
The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph.
In this graph,
On x-axis --- Time. On y-axis --- Velocity.
Velocity-time graph for
1) Uniform acceleration is --- a straight line.
2) Non-uniform acceleration is --- a curve.
3) Uniform velocity is --- a straight line parallel to time-axis.
4) An object at rest is --- a straight line along the time-axis.
Area enclosed by velocity-time graph and the time axis will be equal to --- the distance (magnitude of the displacement) covered by the object.
Slope of the velocity-time graph gives --- acceleration. The acceleration, a of the object, can be represented as
a = (v_{2} - v_{1})/ (t_{2} - t_{1})
Velocity-time Graph of an Object Moving with Uniform Velocity:
If a body moves with a uniform (constant) velocity then velocity time graph for this body would be straight line parallel to time axis.The slope of a velocity–time graph of an object moving in rectilinear motion with uniform velocity is zero.
The product of velocity and time give displacement of an object moving with uniform velocity. The area enclosed by velocity-time graph and the time axis will be equal to the magnitude of the displacement. Calculation of distance using velocity-time graph:
Let two points A and B on the slope of graph. Draw two lines parallel to y-axis AC from point A, and BD from point B. Let point D at the x-axis (time axis) is t_{2} and point C is t_{1}. Let AB meet at ‘v’ at y-axis, i.e. object is moving with a velocity, v.
Thus, distance or displacement by the object is equal to the area of the rectangle (shaded) ABCD.
Thus, Area of ABCD = BD × DC
⇒ s = v (t_{2} – t_{1})
Since given object is moving with constant velocity along a straight line, thus displacement will be equal to distance covered.
Therefore, Distance or Displacement = velocity x time interval.
Velocity – time Graph of an Object Moving with Uniform Acceleration
The velocity time graph of uniformly changing velocity (i.e., uniform acceleration) is a straight line. Slope of velocity time graph of moving body gives its acceleration. The slope of the velocity time graph of an object moving with uniform decreasing velocity with uniform acceleration is a downwards straight line. The straight downward slope shows the decreasing velocity with uniform acceleration, i.e. retardation.
Calculation of Displacement and Distance covered by the moving object using velocity time graph:
The pattern of slope of the graph shows that object is moving with uniform acceleration.
Let take two points, A and B at the slope of the graph. Draw a line from B to BD and another from point A to AE parallel to y-axis. Let AD meets at t_{2} and AE at t_{1} on the time axis.
Thus, Distance covered by the object in the given time interval (t_{2} – t_{1})
is given by the area of ABCDE.
Therefore, Distance (s) = Area of ΔABC + Area of ACDE
⇒ S = (½ × BC × AC) + (AE × ED)
Displacement of the object during the given time interval (t_{2}−t_{1}) = Area of ACDE
Thus, Displcement = AE × ED
Velocity - time Graph of an Object Moving with Non-uniform Velocity:
Zig–zag line of slope of graph shows that the object is moving with non-uniform velocity.