Sample Problems on Acceleration Time Graphs

Problem 1: From the acceleration vs time graph given below, determine the change in velocity.

Solution:

To find the change in velocity of the body we need to determine the area under the curve.

So, to find the change in the velocity of the object, we need to calculate the area of the triangle.

△v = area of triangle

=  ½ × 25 × 6

△v  = 75 m/s.

Hence, the change in velocity is 75 m/s.

Problem 2: From the acceleration vs time graph given below, find the initial velocity of a body if its final velocity is 55 m/sec.

Solution:

To find the change in velocity of the body we need to determine the area under the curve. From the graph, we can observe that it has a rectangle and a triangle. So, to find the change in the velocity of the object, we need to calculate the area of these figures.

△v = area of triangle + area of rectangle 

=  ½ × 8 × 6 + 2 × 8

△v = 24 + 16 = 42 m/s

We know that,

△v = final velocity – initial velocity

42 = 55 – v_i

v_i = 55 – 42 = 13 m/s

Hence, the initial velocity is 13 m/s.

Problem 3: From the acceleration vs time graph given below, determine the velocity at t = 6 sec, if v(0) = 0.

Solution:

Acceleration is given by;

a = dv/dt

⇒ dv = (a)dt

By integrating on both sides, we get

∫dv = ∫(a) dt

⇒ v = ∫(1.5) dt

⇒ v(t) = 1.5t + c, where c is a constant

⇒ v(0) = 0 

⇒ c = 0

Now, v(t) = 1.5t

v(6) = 1.5 × 6 = 9 m/s

Hence, the velocity at t = 6 sec is 9 m/s.

Problem 4: What does the area under the acceleration-time graph indicate?

Solution:

The area under the acceleration-time graph represents the change in velocity. 

Let △v be the change in velocity, △a be the change in acceleration, and △t be the change in time.

Now, the area under the curve = △v

We know that the acceleration of a body is referred to as the ratio of the change in velocity in a given period of time.

So,

△a = △v/△t

Now, by multiplying with △t on both sides, we get,

△v = △a × △t

So, the area under the curve is obtained by multiplying the change in acceleration and change in time. 

Problem 5: What does a jerk in the acceleration time graph mean?

Solution:

A jerk is the sudden change in the acceleration of the moving body, and the slope of the a-t graph represents jerk.

The slope of the acceleration-time graph = jerk = △a/△t

Acceleration is the change of velocity with time. In real life scenario acceleration also changes with time. For example, you are travelling from one city to another city by road. Then in this case you will accelerate more in case of empty road and will accelerate down in case of traffic. This change of acceleration can be observed by plotting acceleration against time in a graph. This called acceleration time graph. In this article, we will learn in detail about acceleration time graph and solve problems based on it.