Velocity-time

The velocity of an object is the rate of change of position of any particle with respect to a frame of reference. Velocity is a function of time. The vertical axis of a velocity-time graph represents the velocity of the object. The horizontal axis represents the time from the start of motion.

Velocity

“The rate at which an object changes its displacement is known as velocity. Velocity is a vector quantity. Consider a person who takes one step forward and one step back at a rapid pace, always returning to the same starting place. While this would result in a flurry of activity, it would also result in a velocity of zero. As the person continually returns to their former posture, the person would never change position. This motion has zero velocity because velocity is defined as the displacement’s rate changes. If a person in motion wishes to accelerate, they must make every effort to increase the displacement between them and their starting point. Every step must be taken to get that person further away from where he or she began. 

Velocity-time graph

Acceleration is shown by the slope of the velocity-time graph. The acceleration is zero if the velocity-time graph has a horizontal slope. This signifies that the object is either stationary or travelling at a consistent pace. When the slope is rising, the rate of acceleration rises. Acceleration decreases if the slope is negative.

The information about the journey taken by an object is provided by velocity-time graphs.

In velocity-time graph, 

On the y axis, velocity is plotted

On the x axis, the time is plotted 

A velocity-time graph can also calculate the displacement travelled by the object and the acceleration of an object.

Imagine a situation where the sprinter is just taking off from the starting blocks for running a short race on a straight track. For the first few seconds of the race, the sprinter will pick up more speed as she starts with a burst of speed. After that, she keeps running at her top speed until she finishes the race by crossing the finishing line. But after that, she will slow down. Acceleration is defined as the rate of change in velocity. In this case, though her direction stays the same, her speed is changing. This means her acceleration will also change.

A velocity-time graph can represent the change in velocity not only of the sprinter but for all moving objects (any kind of object). 

The slope of the velocity-time graph

It is possible to determine the acceleration of an item by examining the slope of a velocity-time graph. If the slope of the velocity-time graph is a horizontal line, then the acceleration is equal to zero. This indicates that the item is either at rest or is travelling at a constant speed. If the slope is positive, the acceleration is positive in magnitude. If the slope is negative, it indicates that the acceleration is negative.. 

Derivation of the equations of motion 

Acceleration is “the rate at which a given velocity changes concerning time,” and in the context of a particular velocity-time graph, it may be expressed as “the slope of the graph.”

a= dvdt = slope of the graph that results from deriving the equations of motion.

⇒ v = u+at…….(1)

The displacement of the particle may be calculated from the area beneath the velocity-time graph.

Now, using the first equation of motion, we may say

(v–u) = at

From the equation of motion, displacement can be given as

s = ut + 12at2 ………..(2)

If we insert the value of ‘t’ from the first equation of motion into the provided equation, we get the following result:

We obtain t=(v–u)/a as a result.

⇒ s = u(v-u)a+12(v-u)2a 

When we solve the equation mentioned above, we get the following result:

⇒ v2 = u2 + 2as …….(3)

Equation (1), (2) and (3) are known as equations of motion.

Conclusion

In this article, we studied velocity-time graphs in various circumstances and scenarios. The equations of motion for constant acceleration were also obtained using the velocity-time graph. We discovered that one of the limitations of these equations of motion is that the acceleration must be constant throughout the equation. The slope of the velocity-time graph determines acceleration.