The acceleration time graph is the graph that is used to determine the change in velocity in the given interval of the time. In the acceleration vs time graph on the x-axis you have the time taken by the object and on the y-axis acceleration of the object, in which the area under the graph gives you the change in velocity of the object over the given period of the time. The acceleration time graph is used to find the change in the velocity of the moving object for the given period of time and this can be determined by finding the area under the curve.

To understand the graph of the acceleration vs time graph you must have an idea about some terminologies. Let us discuss these terminologies first.

Acceleration:- It is the ratio of the change in velocity in the given time interval. The SI unit of the acceleration is m2/sec. In simple words acceleration means to gain and lose the velocity of the vehicle.

Velocity:- The velocity of an object is defined as the ratio of the change in the position of the object in the given time interval. The SI unit of velocity is m/sec.

Time period:- The time period is defined as the interval of the time given to perform the specific activity.

The area under the graph gives the change in the velocity of the object in the given interval of the time. This means the area under the curve of the acceleration time graph is equal to the change in the velocity of the object in the given time period.

The area under curve = △v

You can represent the change in velocity as △v, change in acceleration as △a and change in time as △t

According to the definition of the acceleration,

△a = △v/△t

By multiplying the △t that is change in time interval to both sides, we get,

This is the direct theoretical formula which can be used to find any of the three values for which the other two are given. What if there is only a graph given in the place of the data? In that case, you will figure out the shapes formed under the graph and find the area of that figure according to the parameters required to calculate that.

Let us consider some examples to understand the topic more precisely.

Example:- The initial velocity of an object is 6m/sec, the change in the acceleration of the object in 10 seconds is 2m2/sec. Find the final velocity of the object?

Answer:- As there is the direct data given in the question so here solve the question by the formula that we have,

△v = △a△t

△a that is a change in acceleration = 2m2/sec, △t which is symboled for the time interval = 10 sec so by the formula we get the change in velocity as,

△v = △a△t

△v = 2 x 10

△v = 20 m/sec

△v is the change in velocity which means that the difference between the final velocity and initial velocity of the object.

△v = vf - vi

20 = vf - 6 ( initial velocity is given in the question as 6 m/sec)

vf = 26 m/sec

The final velocity of the object is 26 m/sec.

Example:- There is an acceleration vs time graph given, then find the initial velocity of the object if the final velocity of the object was 40m/sec.

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Answer:- As there is a graph that has the figure under the curve so you have to find the area under the acceleration time graph.

As you can see clearly, the graph has three figures, two triangles and one rectangle. So you need to find the area of these figures to find the change in the velocity of the object.

△v = area of rectangle + area of △1+ area of △2

△v = LB+ ½ base.height+½ base.height (data as per the given graph)

△v = 3x4 + ½ x 4x 4 + ½ x 2 x 2

△v = 12 + 8 + 2

△v = 22 m/sec

△v is the change in velocity which means that the difference between the final velocity and initial velocity of the object.

△v = vf - vi

22 = 40 - vi ( final velocity of the object 40 m/sec given in the question)

vi = 40-22

vi = 18m/sec

The final velocity of the object is 18m/sec.

FAQ (Frequently Asked Questions)

1. How can you find the area under the acceleration time graph having an uneven curve?

The area under the acceleration time graph is the sum of the small rectangles which can be formed under any kind of curve. It is obvious that the area of a rectangle and a curve will give you the different answers but when you divide that curve into the smallest rectangle which gives you the negligible difference in the area that would not make any considerable change in the answer. And to find the area under the curve break that curve into those figures whose area is easy to find.

2. Can we find the velocity of the object at a particular point of time with the help of an acceleration time graph?

No. One cannot calculate the velocity of the object directly from the graph at a particular time. However, there is another way to find it. We can only find the change in velocity by the area under the acceleration time graph by calculating the area under the curve to the given time. Then by the formula of the difference in the final and initial velocity you can find the final velocity of the object in the given time period.