Motion in One Dimension
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
6.0 Analysis of motion through graph
6.2 Velocity - time graph
6.3 Area under the graph
6.4 Solved examples
The graph helps to analyze and visualize the motion of a body.
Before we start graph, let us understand basic terminologies of the graph.
Slope $\left( {\frac{{dy}}{{dx}}} \right)$
$y=mx+c$ be a straight line.
$m$ is the value of slope
Slope means the inclination of line with respect to the $x$ axis.
$$\tan \theta = m = \frac{{dy}}{{dx}}$$
The first derivative gives the slope.
Note:
1. The slope of a straight line is constant.
2. The slope of a curve is different at a different point.
For finding slope at any point, draw a tangent at that point. The angle made by the tangent with the $x$ axis gives the slope at that point/
Slope at point $A$: ${m_A} = \tan {\theta _1}$
Slope at point $B$: ${m_B} = \tan {\theta _2}$
Slope at point $C$: ${m_C} = \tan {\theta _3}$
The graph helps to analyze and visualize the motion of a body.
Before we start graph, let us understand basic terminologies of the graph.
Slope $\left( {\frac{{dy}}{{dx}}} \right)$
$y=mx+c$ be a straight line.
$m$ is the value of slope
Slope means the inclination of line with respect to the $x$ axis.
$$\tan \theta = m = \frac{{dy}}{{dx}}$$
The first derivative gives the slope.
Note:
1. The slope of a straight line is constant.
2. The slope of a curve is different at a different point.
For finding slope at any point, draw a tangent at that point. The angle made by the tangent with the $x$ axis gives the slope at that point/
Slope at point $A$: ${m_A} = \tan {\theta _1}$
Slope at point $B$: ${m_B} = \tan {\theta _2}$
Slope at point $C$: ${m_C} = \tan {\theta _3}$
Nature of the curve $\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)$
Second derivative tells us the nature of the curve.
| Condition | Nature | Graph | Equation | Derivation |
| $$\frac{{{d^2}y}}{{d{x^2}}} = 0$$ | Straight line |
| $$y = mx + c$$ | $$\frac{{dy}}{{dx}} = m$$$$\frac{{{d^2}y}}{{d{x^2}}} = 0$$ |
| $$\frac{{{d^2}y}}{{d{x^2}}} > 0$$ | Concave up |
| $$y = {x^2} + 5$$ | $$\frac{{dy}}{{dx}} = 2x$$$$\frac{{{d^2}y}}{{d{x^2}}} = 2>0$$ |
| $$\frac{{{d^2}y}}{{d{x^2}}} < 0$$ | Concave down |
| $$y = -{x^2} + 5$$ | $$\frac{{dy}}{{dx}} = -2x$$$$\frac{{{d^2}y}}{{d{x^2}}} = -2<0$$ |
