Acceleration is defined as change in velocity per unit time. It is a vector quantity and Average acceleration formula is defined as

$a = \frac {Final \; velocity – Initial \; velocity}{ time \; taken }$

We will look at steps on how to find acceleration with velocity and time in various situation

__Uniformly Acceleration Motion__

Acceleration for a object moving in straight line with constant acceleration is defined by

$a = \frac {v- u}{ t}$

Where v is final velocity and u is initial velocity and t is time taken

**Example 1**

A object start from rest and acquire the velocity 5 m/s in time 10 sec with constant acceleration. What is the acceleration?

**Solution**

We have v=5 m/s and u=0 (as it rest)

$a = \frac {v- u}{ t}$

$a = \frac {5- 0}{ 10} = .5 m/s^2$

__Uniformly Deceleration Motion__

Deceleration for a object moving in straight line with constant deceleration is defined by

$a = \frac {v- u}{ t}$

Where v is final velocity and u is initial velocity and t is time taken. The value will have negative sign

**Example 1**

A car comes to rest from the velocity 10 m/s in time 10 sec with constant retardation. What is the Deceleration ?

**Solution**

We have u=10 m/s and v=0 (as it rest)

$a = \frac {v- u}{ t}$

$a = \frac {0- 10}{ 10} = – 1 m/s^2$

__Motion in a straight line with non-uniform acceleration__

We define average acceleration in a time interval as

$a = \frac {v_2 – v_1}{ t}$

Where $v_2$ and $v_1$ are final and initial velocities

**Example 3**

A car accelerates from the velocity 10 m/s to 30 m/s in 2 sec . What is the average acceleration of the car in that interval ?

**Solution**

We have $v_2=30$ m/s and $v_1=10$ m/s

$a = \frac {v_2 – v_1}{ t}$

$= \frac {30- 10}{ 2} = 15 m/s^2$

__Acceleration Motion in a plane__

Acceleration in plane in vector form is defined as

$\boldsymbol{\mathbf{a}}=\frac {\mathbf{v_2} -\mathbf{v_1} }{t}$

where $\mathbf{v_2}$ and $\mathbf{v_1}$ is the velocity vector and t is the time taken

**Example 4**

A object accelerates from the velocity 2**i** + 3**j** to 10**i** + 3**j** in 2 sec . What is the average acceleration of the object in that interval Here **i** and** j **are unit vectors across x and y axis of the Cartesian plane ?

**Solution**

$\boldsymbol{\mathbf{a}}=\frac {\mathbf{v_2} -\mathbf{v_1} }{t}$

$\boldsymbol{\mathbf{a}}=\frac {10\mathbf{i} + 3\mathbf{j} -2\mathbf{i} -3 \mathbf{j}}{2}

=4\mathbf{i}$

__How to find Instantaneous acceleration from velocity and time using Calculus__

Instantaneous acceleration is defined as

$a = \frac {dv}{dt}$

we can use above formula to calculate instantaneous acceleration when velocity is expressed as function of time

**Example 5 **

A object moves on x axis such that velocity varies with time as

$v= 3 + t + 2t^2$ m/s

Find the instantaneous acceleration as a function of time and instantaneous acceleration at t=0

**Solution**

$v= 3 + t + 2t^2$ m/s

Now

$a = \frac {dv}{dt}$

So

$ a= \frac {d}{dt} 3 +t + 2t^2 $

$a= 1+ 4 t$

Now at t=0

a =1 m/s^{2}

**Practice Questions based on the above scenario’s**

**Question 1**

A particle is moving up an inclined plane. Its velocity changes from 12 m/s to 10 m/s in two seconds. What is its acceleration?

**Question 2**

The velocity changes from 45 m/s to 60 m/s in Three seconds. What is its acceleration?

**Question 3**

A stone is thrown in a vertically upward direction with a velocity of 10 m/s. If the acceleration of the stone during its motion is 10 m /s^{2} in the downward direction, what will be the height attained by the stone and how much time will it take to reach there?

**Question 4**

Velocity of the particle in a straight line varies with time as defined below

$v= 3t + 5t^2 + 10$

a. What is acceleration at t=0 sec

b. What is the acceleration at t =1 sec

Hope you like these examples,explanation steps on how to find acceleration with velocity and time. Please do provide the feedback

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