Average Acceleration – Definition

The change in velocity divided by the elapsed time is the average acceleration. For example, if a marble’s velocity increases from 0 to 60 cm/s in three seconds, its average acceleration is 20 cm/s. This indicates that every second, the marble’s velocity will rise by 20 cm/s.

Acceleration for an object is defined as the rate of change of velocity.

Average acceleration for an interval is defined as the change in velocity for that interval per time

So, mathematically,

acceleration(a) =

△v△t△�△�

If v_i is the initial velocity and v_f is the final velocity, and t is the time taken for the velocity to change from v_i to v_f, then

a=vf−vtt�=��−���

Similarly, when the object executes various velocities, such as v₁, v₂, v₃…v_n for varying time intervals such as t₁, t₂, t₃…t_n respectively, the average acceleration formula is written as:

=

aavg=v1+v2+…..+vnt1+t2+…..+tn����=�1+�2+…..+���1+�2+…..+��

SI unit of average acceleration is . m/s². 

Examples:

1. If an object accelerates from 20 m/s to 80 m/s in 3 seconds, Find the average acceleration for the object. 

Sol:

Given,

The initial velocity of the object=vi=20 m/s

The final velocity of the object=vf=80 m/s

The time taken by the object=t= 3 seconds

Now, we are asked to calculate the average acceleration of the object. We know that the average velocity of the object can be calculated using the following equation:

aavg=vf−vit����=��−���

…..(1)

Where,

vi = The initial velocity of the object

vf  = The final velocity of the object

t-The time taken by the object

Substituting all the known values in the above equation we get:

=

aavg=vf−vit=80−203����=��−���=80−203

=

aavg=403=13.33m/s2≈13m/s2����=403=13.33�/�2≈13�/�2

Therefore, the average velocity of the object is13 m/s2.

2. A bus accelerates for 5 seconds at 10 mph, then 20 mph for 4 seconds, then 15 mph for 8 seconds. What can we say about the bus’s average acceleration?

Sol:

It is assumed that the bus’s velocities at certain time intervals are, v₁ = 10 m/s, v₂ = 20m/s, v₃ = 15m/s

The Time intervals for which the item has these velocities  are, t₁ = 5s, t₂ = 4s, t₃ = 8s

As a result, the overall velocity across the interval can be calculated as the sum of these velocities.

V_total = v₁+v₂+v₃ = 10+20+15 = 45m/s

Similarly, the total time interval can be calculated by adding these intervals together.

t_total = t₁+t₂+t₃ = 5+4+8 = 17s

Now, we are asked to determine the bus’s average acceleration. The average acceleration formula is given by:

=

aavg=v1+v2+…..+vnt1+t2+…..+tn����=�1+�2+…..+���1+�2+…..+��

…..(1)

aavg=4517=2.647m/s2≈2.65m/s2����=4517=2.647�/�2≈2.65�/�2

Therefore, the average velocity of the bus is 2.65 m/s².

This is how the average acceleration formula is derived and used to solve problems. Understand the use of different terms and their implications in the formula in the solved examples. 

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