Velocity-Time Graph

Traveling in vehicles has become an important part in our daily lives

Say, we have a truck which moves in a straight line from A to B, over a distance of 100 km in 4 hrs

We know that Velocity = $\frac{Displacement}{Time}$ Since the truck is travelling in a straight line, displacement = 100km

Hence, the velocity = $\frac{100km}{4hrs}=25km/hr$ But, we must remember that this is only the average velocity

It is possible that the truck is not moving the entire time with a velocity of $25km/hr$

It can encounter a traffic signal, say at point C, where it has to slow down to rest

Or it can increase it's speed up on a highway

So, the truck may not have constant velocity throughout it's journey. It may have to speed up or come to rest depending on different situations

Such a motion, where the velocity of a moving vehicle is changing with time, is called as "accelerated motion"

We define acceleration to be the rate of change of velocity

Acceleration of an object is defined to be the rate of change of velocity of that object

Mathematically, $\text{Acceleration}=\frac{\text{Change in Velocity}}{\text{Change in Time}}$

If the velocity increases with time, then the object is said to have "Positive Acceleration", while if the velocity decreases, it is called "Deceleration"

Let us try to plot the velocity and time on a graph

On a graph, we plot $Velocity$ on the $Y$ Axis and $Time$ on the $X$ axis to get the Velocity-Time Graph

Case 1: Let us assume that the vehicle is moving with a constant velocity of $50km/hr$ over a time interval of $0-5hrs$

The Velocity-Time Graph would look something like this

The area under the Velocity-Time Graph gives the Displacement in the time interval

We can prove it analytically as: $Displacement=Velocity\times Time$ $⟹$ Displacement after 5 hrs = $50\times 5=250km$

From the graph area under the curve = area of rectangle AOCD = $50\times 5=250km$

The slope of the Velocity-Time Graph gives the value of Acceleration

Since the vehicle is moving with a constant velocity, the acceleration = 0

We can also say from the graph since the velocity plot is parallel to the time

Case 2: Let us assume the vehicle is moving with a velocity starting from rest, that is constantly increasing over a time interval of $0-5hrs$

The Velocity Graph would look something like this

On similar lines, the displacement can be obtained from the area under the curve

Displacement $=\frac{1}{2}(50)(5)=125km$

Here, the acceleration is not $0$, since the velocity is constantly changing with time

Acceleration $=\frac{50-0}{5-0}=10km/h{r}^2$ The velocity increases by $10km/hr$ every $1hr$

Case 3: Let us assume that the vehicle is moving with a velocity that is constantly decreasing over a time interval of $0-5hrs$ and finally coming to rest

The Velocity-Time graph would look something like this

Displacement = Area under the curve = $\frac{1}{2}(50)(5)=125km$

The Acceleration is negative in this case, since the velocity is constantly decreasing over time

Deceleration $=\frac{0-50}{5-0}=-10km/h{r}^2$ (Since the initial velocity is taken to be $50km/hr$)

The velocity is decreasing by $10km/hr$ every $1hr$

Revision

Average Velocity = (Total Displacement)/(Total time) Acceleration = (Change in velocity)/(Change in time interval)

Velocity-time graph is a graphical representation of the motion of an object with the Velocity on the Y-Axis and Time on the X-Axis

For constant velocity, the $v-t$ curve is a horizontal straight line. For uniformly accelerated motion, its a straight line with positive slope

For uniformly decelerated motion, its a straight line with negative slope

The End