# Average Acceleration Formula

Acceleration: Acceleration of an object is defined as its rate of change of velocity at that instant. It is usually denoted by ‘a’ and is measured in the units of m/s2. Any object experiencing an unbalanced force, constant or variable, shows acceleration. In other words, any object undergoing a change in its velocity, either magnitude, or direction or both is said to be accelerated.

Average acceleration: Average acceleration is defined as the ratio of change in velocity to the change in time for a given interval. Unlike instantaneous acceleration, average acceleration is calculated for a given interval.

Formula:

Average acceleration is calculated by the following formula,

$Average Acceleration = \Delta _{v}/\Delta _{t}$

Here, Δ v is the change in velocity and Δ t is the total time over which the velocity is changing.

Here the change in velocity and time can be expressed as the difference in the initial and the final velocity. Let us say, the initial time is denoted by ti, and the velocity at this time is observed to be vi. After some time it attains the final velocity vf when the time is tf. The above expression for average acceleration can be written as

$Average Acceleration = v_{f} - v_{i}/ t_{f} - t_{i}$

Here, the initial velocity is vi, the final velocity is vf, the initial time is ti and the final time is tf.

Also, if the object shows different velocities, such as v1, v2, v3…vn for different time intervals such as t1, t2, t3…t3 respectively, the average acceleration is calculated using the following formula,

$Average Acceleration = v_{1}+ v_{2}+ v_{3}+.....+v_{n}/ t_{1}+ t_{2}+ t_{3}+....+t_{n}$

Real Life Example

If the velocity of a marble increases from 0 to 60 cm/s in 3 seconds, its average acceleration would be 20 cm/s/s. Meaning that the marble’s velocity will go up by 20 cm/s each second.

Average acceleration Problems

Example 1: A bus accelerates with an initial velocity of 10 m/s for 5s then 20 m/s for 4s finally for 15 m/s for 8s. What can be said about the average acceleration of the bus?
Solution:

It is given that, the velocities of the bus at different time intervals is, v1 = 10 m/s, v2 = 20m/s, v3 = 15m/s
The time intervals for which the object possesses these velocities are t1 = 5s, t2 = 4s, t3 = 8s
Hence, over the interval, the total velocity can be given as the sum of these velocities.

$\Delta v = 10+20+15=45 \frac{m}{s}$

Similarly, the total time interval can be given as the sum of these intervals,

$\Delta t = t _{1}+ t_{2}+ t_{3}= 5+4+8 = 17s$

Using the above formula for average acceleration, we get,

$Average Acceleration = \Delta _{v}/\Delta _{t}$

$Average acceleration = \frac{45}{17} = 2.65 \frac{m}{s^{2}}$

Question 2: A sparrow, while going back to its nest accelerates to 6 m/s from 3 m/s in 5s. What can we say about its average acceleration?
Solution:

As given in the question, the initial velocity vi = 3m/s,
And the final velocity vf = 6m/s
Also, the total time interval over which the velocity changed can be given as, t = 5 s

Using the above mentioned formula for average acceleration, we get,

$Average Acceleration = v_{f} - v_{i}/ t_{f} - t_{i}$

$Average acceleration = \frac{6-3}{5} = 0.6 \frac{m}{s^{2}}$