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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 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/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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