## Graphical Representation Of Motion

**Displacement-time Graph:** A graph obtained by plotting t (time) along the x-axis and s, the corresponding distances travelled by a particle along the y-axis is called a distance-time graph. When the corresponding distances (s) are plotted as displacements along the y-axis, the graph is called a displacement-time graph. These graphs represent the changes in position and, hence, the displacement of a particle with time.

Graphs for a particle

- At rest,
- In motion with uniform velocity,
- In motion with a uniform acceleration and
- In motion with non-uniform acceleration

The point P denotes the displacement OR of the particle in time OQ. In this figure, the gradient of the displacement-time graph determines the velocity of the particle. The straight line has a uniform gradient—so the velocity is uniform.

**Read and Learn More: Class 11 Physics Notes**

According to the particle is displaced by s_{2} – s_{1} = CB in the time interval t_{2} – t_{1} = AC.

Therefore, the average velocity of the particle in that interval = of time \(v=\frac{s_2-s_1}{t_2-t_1}=\frac{C B}{A C}=\) gradient of the chord AB.

To find the instantaneous velocity of the particle at time t_{1}, the time interval (t_{2 }– t_{1}) needs to be infinitesimally small. Hence, the point B is almost superimposed on point A.

In this condition, the gradient of the chord AB becomes equal to the gradient of the tangent drawn at A. Thus, the gradient of the tangent drawn at any point on the displacement-time graph denotes the instantaneous velocity of the particle at the corresponding moment.

**Velocity-time Graph:** A velocity-time graph is drawn by plotting time t along the x-axis and velocity v along the y-axis. Show velocity-time graphs for a particle

- Moving with a uniform velocity,
- Starting from rest and moving with a uniform acceleration,
- Starting with an initial velocity of u and accelerating uniformly and
- In motion with non-uniform acceleration

The point B denotes the magnitude of the velocity OC of the particle in time OA. In these figures, the gradient of the velocity-time graph gives the acceleration of the particle. The gradient of CB is zero—so there is no acceleration. But the gradient of OB is positive and uniform—so the acceleration is uniform.

- The area under the velocity time graph and the time axis gives the displacement of the particle.
- The average acceleration of the particle in the time interval (t
_{2}– t_{1}) is equal to the gradient of the chord AB. The graph denotes the motion of a particle moving with non-uniform acceleration. - With the help of calculus, it can be shown that for a particle moving with a non-uniform acceleration, its displacement for any interval of time is equal to the area enclosed by the arc denoting the motion, the time interval and the time axis.