Acceleration time graph-

The acceleration time graph depicts the relation between acceleration and time for a particle moving in a straight line. Acceleration vs time graph is an effective method to determine the change in velocity over a time period.

How to plot the Acceleration time graph?

• In the acceleration graph, the independent variable- acceleration is plotted on the y-axis, and the dependable variable- time is plotted on the x-axis.

• Acceleration ( independent variable) ( unit = m/sec^2) in the y-axis.
• Time ( dependent variable) ( unit = seconds) in the x-axis.

What does the slope of the Acceleration time graph represent?

• The slope of the acceleration time graph represents – Jerk
• Jerk (symbol j) is the rate of change of acceleration with respect to time. The SI unit of jerk is m/s³.

What does the area under the Acceleration time graph signify?

• The area under the acceleration vs time graph represents the change in velocity ( ∆v ). In other words, the change in velocity in a particular interval of time is the area under the acceleration- time graph for the same interval of time.
• Area = ∆v
• Derivation-

As we all know,

Area under the graph (A)= ∫a.dt (equation 1)

And a= dv/dt

So, dv= a.dt (equation 2)

From equation-1 and equation-2 we get –

A= ∫dv ( integrating from v1 and v2)

A= v2 -v1

A= Δv

Where, A= area under the graph

a = acceleration

Δv= change in velocity

Δt= change in time

Examples ( Acceleration- Time graph)

Example 1

Q- Determine velocity at t= 6 seconds if the velocity at t= 0 was 0 using the acceleration time graph given below.

A- as we all know

a= dv/dt

dv= a.dt

Integrating both the sides

∫dv= ∫a,dt

v= ∫(1,5)dt

v(t)= 1.5t + c

Where C is a constant

v (at t=0)= 0

Therefore, c=0

v(t)= 1.5t

v( at t=6)= 1.5 X 6

v= 9 m/s ( at t= 6 seconds)

Example 2

Q- The initial velocity of a truck is 20m/sec at time 0 seconds, the time acceleration graph below shows the acceleration of the truck. Find the final velocity (vf) of the truck.

A-

To find out the change in velocity we will calculate the area under the curve

• Area = ∆v = (½) (6 m/s^2) ( 8 sec)

∆v = 24m/s

As we all know that the change in velocity is the final velocity minus the initial velocity

Δv= vf- vi

Putting in the values

24 m/sec = vf- 20 m/sec

vf= 44 m/sec

The final velocity of the truck will be = 44m/sec

Conclusion –

The acceleration time graph depicts the relation between acceleration and time for a particle moving in a straight line. The area under the acceleration time graph will give us a change in velocity, to calculate the change in velocity between two points you have to connect those two points with the x-axis using vertical lines and then find out the area of the newly formed quadrilateral/ triangle. Also, the slope of acceleration vs time graph gives a quantity known as a jerk. A jerk is a change in acceleration of any object, in simple words jerk is a sudden change in acceleration that generally lasts for a small period. One example is the palm of a bowler while bowling in cricket. Convert the given quantities like velocity, acceleration, and time into the correct units before applying them in a formula.

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