Average Acceleration Formula

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

Understanding acceleration is important in our everyday lives, as well as in the wide expanses of space and the microscopic world of subatomic physics. Applying the brake pedal causes a vehicle to slow down; in ordinary conversation, to accelerate indicates to hurry up. For example, we are all too familiar with our car's acceleration. The velocity change over time is proportional to the acceleration. 

Acceleration is a term you've probably heard before, especially in relation to automobiles. As a result, acceleration is defined as the rate at which velocity changes over time. We use the term acceleration to express a situation of increased speed in layman's terms. Acceleration is divided into two types: average and instantaneous, similar to velocity. Over a long period, the average acceleration is calculated. We use the term length in the sense of something finite, i.e. something that has a beginning and an end. As a result, the focus of this paper will be on the average acceleration formula.

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) = \[\frac{\triangle v}{\triangle t}\]

If vi is the initial velocity and vf is the final velocity, and t is the time taken for the velocity to change from vi to vf, then \[a=\frac{v_{f}-v_{t}}{t}\]

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

= \[a_{avg}=\frac{v_{1}+v_{2}+.....+v_{n}}{t_{1}+t_{2}+.....+t_{n}}\]

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:

\[a_{avg}=\frac{v_{f}-v_{i}}{t}\] …..(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:

= \[a_{avg}=\frac{v_{f}-v_{i}}{t}=\frac{80-20}{3}\]

= \[a_{avg}=\frac{40}{3}= 13.33m/s^{2}\approx 13 m/s^{2}\]

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.

Vtotal = v1+v2+v3 = 10+20+15 = 45m/s

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

ttotal = 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:

= \[a_{avg}=\frac{v_{1}+v_{2}+.....+v_{n}}{t_{1}+t_{2}+.....+t_{n}}\]…..(1)

\[a_{avg}=\frac{45}{17}= 2.647m/s^{2}\approx 2.65 m/s^{2}\]

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.