# how to find acceleration with velocity and time

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  2i + 3j  to 10i + 3j in 2 sec . What is the average acceleration of the object in that interval Here i andare 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/s2

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 /s2 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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