# 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

Related Articles

Acceleration formula Explained with Examples

What is Acceleration of free fall

Acceleration in a curvilinear motion

How to effectively solve Motion in One dimension Problems

Acceleration calculator