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
$Change in the x-variableChange in the y-variable $
Thus, the slope of the curve is given,
$dxdy =dtdv =a$
which is the acceleration
Thus the slope of the curve is,
$5/60−0hr60−0km/h =hr_{2}720km =a$
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,
$∴$
$a=dtdv =0$
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
$∞$
negative acceleration (Deceleration)
The acceleration is -
$∞$
, because slope becomes -
$∞$
as
$dt$
in
$dtdv $
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
$Change in the x-variableChange in the y-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
$∞$
slope) in Velocity-time graph indicates collision like events, with a very sudden change in velocity
The End