# A Particle Starts from Rest from Origin and It’S Acceleration is Given by a = K/(X+4)^2 M⁄S^2.Knowing that V = 4 M/S When X = 8m,Find :(1)Value of K(2)Position When V = 4.5 M/S - Engineering Mechanics

A particle starts from rest from origin and it’s acceleration is given by a = k/(x+4)^2     m⁄s^2.Knowing that v = 4 m/s when x = 8m,find :
(1)Value of k
(2)Position when v = 4.5 m/s

Given :  Particle starts from rest
a = k/(x+4)^2    m⁄s^2
v = 4m/s at x = 8m
To find : Value of k and position when v= 4.5m/s

#### Solution

Solution:
We know that a = v(dv)/(dx)

v(dv)/(dx)=k/((x+4)^2

v dv = k(x+4)-2 dx
Integrating both sides

∫𝑣𝑑𝑣 =∫k(x+4) −2 dx
(v^2)/ 2=(−k)/(x+4 )+ c1 ………(1)
Putting x=0 and v=0
c1=k/4  ……….(2)
(v^2)/2 = (−k)/(x+4 )+ k/4   …….(From 1 and 2) ……..(3)
k = 48
From (3)
(v^2)/2 =( −48)/(x+4)  + 48/4

v^2=24 - 96/(x+4)
Substituting v=4.5 m/s
4.5^2 = 24 - 96/(x+4)
96/(3.75) = x+4
x = 21.6 m
Value of k = 48
The particle is at a distance of 21.6 m from origin when v = 4.5 m/ s

Concept: Introduction to Basic Concepts
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