Velocity-Time Graph I

Graphs are important, to see how things change, with respect to one another

One such graph can be how many electric vehicles are sold over the years

This graph for example tells us sales have been increasing

Similarly graphs can be also used to understand motion better

In the study of body's motion, velocity is one quantity which is plotted against time

It can be used to pictorially represent how a car changes its speed during the traffic

Since velocity changes with respect to time, it is a function of time

Thus, in a velocity-time graph, velocity is on the y-axis and time is on the x-axis

Let us learn velocity-time graphs through an example

This is a car, whose velocity we will track using the velocity-time graph

The car started with a speed which was increased to 60 km/h in 5 minutes. We will plot the graph of velocity as a function of time

The velocity is plotted in the y-axis and the time in the x-axis

The Velocity-Time graph will be a straight line passing through the Origin

The slope of a graph is given by $\frac{\text{Change in the y-variable}}{\text{Change in the x-variable}}$

Thus, the slope of the curve is given, $$\begin{gathered} \frac{dy}{dx}= \\ \frac{dv}{dt}=\text{a} \end{gathered}$$ which is the acceleration

Thus the slope of the curve is, $$\begin{gathered} \frac{60-0km/h}{5/60-0hr}= \\ \frac{720km}{h{r}^2}=\text{a} \end{gathered}$$ which is the acceleration

Now, the car for the next 10 minutes is travelling east with a uniform speed of 60 km/h. Let us see how the graph would be

Now, since the car is travelling at a constant velocity for the duration of 10 minutes, then, the plot is a line parallel to the x-axis

Now, for a line parallel to the x-axis, the slope = 0, $\therefore$$\text{a}=\frac{dv}{dt}=0$ from 5 to 15 minutes

Now, the driver decides to race the car for the next 5 minutes, such that the velocity increases non-uniformly

Now, for a non-uniform change in velocity over time, we have the Velocity-Time is a "Curve"

Here, therefore, there can be different accelerations at different time, given by different tangents to the curve

Also, without knowing the mathematical function of our velocity here, we cannot find the acceleration at any instant

Now, let us consider a situation where the car undergoes a collision and stops, and see how its graph looks like

If our collision happens in a very short time the velocity goes to zero almost with $\infty$ negative acceleration (Deceleration)

The acceleration is -$\infty$, because slope becomes -$\infty$ as $dt$ in $\frac{dv}{dt}$ is very small here

Thus, driving at high speeds we are more prone to collisions! Drive safe!

Revision

Velocity is the speed with direction and is a function of time

In a Velocity-time graph, velocity is taken along y-axis and time in the x-axis

The slope of a graph is given by $\frac{\text{Change in the y-variable}}{\text{Change in the x-variable}}$

The slope of the velocity-time graph gives us acceleration

When velocity's function is given we can differentiate with respect to time, to get the acceleration as a function of time

When acceleration is non-uniform with velocity function as unknown, acceleration cannot be found

Very steep lines (near $\infty$ slope) in Velocity-time graph indicates collision like events, with a very sudden change in velocity

The End