Uniformly Accelerated Motion (Graphical Treatment)

he velocity-time graph for uniformly accelerated rectilinear motion (motion along a straight line), a > 0, is illustrated in the figure.

The velocity-time graph for a uniformly retarded motion is as shown in the figure. Here a < 0, indicating retardation or deceleration.

In this section, we are going to derive the velocity-time relation for a particle moving with constant acceleration.

The figure above illustrates a particle moving with constant acceleration 'a' along the positive direction of X-axis. If v (t₁) and v (t₂) be the velocities of the particle at times t₁ and t₂ respectively, then

Acceleration, according to the definition

Cross-multiplying, we have,

v (t₂) - v (t₁) = a (t₂ - t₁)

 v (t₂) = v (t₁) + a (t₂ - t₁)

If the motion is considered from t = 0, then the corresponding velocity v(0) at t = 0 is called the initial velocity and the velocity after a time interval, say t, is called the final velocity and denoted by v(t). In this case, the acceleration is given by

or at = v (t) - v (0)

v(t) = v (0) + at

The above expression represents the first equation of motion.

The figure below illustrates the velocity-time graph AB of a uniformly accelerated particle. The points A and B correspond to velocities v (0) and v (t) respectively.

Slope of the straight line AB

Also, from the definition of acceleration, we know that

 We conclude that the slope of the velocity-time graph for uniformly accelerated motion gives the acceleration.At any given time, the direction of motion is given by velocity and not by acceleration. As an example, when a ball is thrown up vertically, its velocity is directed upwards at any given time, but its acceleration is always directed in the downward direction.