Physics > Motion in One Dimension > 6.0 Analysis of motion through graph
Motion in One Dimension
1.0 Introduction
2.0 Kinematic variables
2.1 Distance and displacement
2.2 Average speed and velocity
2.3 Instantaneous speed and velocity
2.4 Average and instantaneous acceleration
3.0 Motion in one dimension
3.1 Motion in a straight line with uniform velocity
3.2 Motion in a straight line with uniform acceleration
3.3 Motion in a straight line with non-uniform acceleration
4.0 Derivation of the kinematics equation
5.0 Vertical motion under gravity
5.1 Basic terminologies for motion under gravity
5.2 Detailed concept of motion under gravity
5.3 Solved examples
6.0 Analysis of motion through graph
6.1 Displacement - time graph
6.2 Velocity - time graph
6.3 Area under the graph
6.4 Solved examples
7.0 Relative motion
7.1 Relative displacement
7.2 Relative velocity
7.3 Relative acceleration
7.4 Illustration of relative motion
7.5 Application of relative motion
8.0 Simultaneous motion of two bodies
9.0 River boat problem
9.1 Downstream
9.2 Upstream
9.3 Crosses the river in shortest interval of time
9.4 Reaches the point just opposite from where he started
9.5 River-man problem
9.6 Solved examples
10.0 Aircraft-wind problem
11.0 Rain problem
6.3 Area under the graph
2.2 Average speed and velocity
2.3 Instantaneous speed and velocity
2.4 Average and instantaneous acceleration
3.2 Motion in a straight line with uniform acceleration
3.3 Motion in a straight line with non-uniform acceleration
5.2 Detailed concept of motion under gravity
5.3 Solved examples
6.2 Velocity - time graph
6.3 Area under the graph
6.4 Solved examples
7.2 Relative velocity
7.3 Relative acceleration
7.4 Illustration of relative motion
7.5 Application of relative motion
9.2 Upstream
9.3 Crosses the river in shortest interval of time
9.4 Reaches the point just opposite from where he started
9.5 River-man problem
9.6 Solved examples
The area under the graph is the product of the horizontal axis and the vertical axis.
Area under $v-t$ graph
The product of velocity and time gives displacement.
So, the area under the graph will give displacement.
Area $ = \frac{1}{2}ab$
So,
Similarly area under speed$-$time graph gives distance traveled.
Displacement $= \frac{1}{2}ab$
Similarly area under speed$-$time graph gives distance traveled.
Note: Area under velocity$-$time graph gives displacement as well as distance.
Illustration:
As velocity is a vector quantity it can be both positive and negative.
Negative velocity means velocity is in opposite direction.
$A_1, \ A_2$ and $A_3$ are the areas of different section.
$A_1>0$ above the time axis
$A_2<0$ below the time axis
$A_2<0$ below the time axis
$A_3>0$ above the time axis
Displacement $=A_1+A_2+A_3$
When area under $v-t$ graph is added with proper sign, it gives displacement.
In other words, vector sum of area gives displacement.
Distance $ = \left| {{A_1}} \right| + \left| {{A_2}} \right| + \left| {{A_3}} \right|$
Scalar sum of area under $(v-t)$ graph gives displacement.
