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Wednesday, 7 March 2018

Know accelerated motion concept

Hello Friends,

Know accelerated motion concept

This topic is important for motion of an object in different state and we will study motion equation in this topic lets start.
if a body moves from one position to other position in such a way that its velocity chances every time this type of motion is called accelerated motion now velocity is displacement per unit time here velocity may be increasing or decreasing if increasing it is called accelerated motion if decreasing it is called decelerated or retardation  motion.





acceleration is define as rate of change of velocity see picture below.          
                                               
Know accelerated motion concept
Know accelerated motion concept
   
here from above picture it is clear that at t =1s velocity is 5m/s and at t = 2s velocity increases to 10m/s and t = 3s velocity is 15m/s and keeping increasing hence change in velocity is (v₂ - v₁)/(t₂ - t₁) so a = (v₂ - v₁)/(t₂ - t₁) when body moves in a straight line called rectilinear motion .
u = initial velocity 
A→→→→→→→→→→→→→→B
t = 0                        s

here u is initial velocity and t is time when object start moving in straight line AB when object start moving two important parameter are changing one is time and other is position of object which is also called displacement so there are two variables changing suppose object start moving from point A when time start t =0 now after t second object reaches at point B also if at A time was t₁ and after reaching B time was t₂ then total time taken will be t₂ - t₁ = t where t is time period remember time period is duration of time now the displacement becomes A to B is s m now position of object at point A is x₁ and position at point B is x₂ then displacement will be final position - initial position = x₂ - x₁ = s we have already study about displacement topic. now suppose velocity is changing from point A velocity is u and at point B final velocity became v hence change in velocity in t time will be a = (v - u)/t now arrange 
at = v - u or v = u +at  where a is acceleration this becomes first equation of our kinematics hence this equation state that if a body start moving with initial velocity u and acceleration a then its final velocity will be  v = u + at  now acceleration unit m/s₂  dimension LT⁻²  mass has no effect on acceleration now whether it is a scalar or vector quantity answer is it is a vector quantity why ? because it has magnitude as well as direction because change of velocity per unit time is acceleration now important is direction of acceleration it is some tricky so we take example below .
u = 5 m/s                                    v = 8 m/s  
A→→→→→→→→→→→→B 

time taken A to B is 2 s now acceleration is vector quantity hence we can write as 
→    →   →
a =   ( v -  u)/2  = a  = (8-5)/2 = 1.5 m/s₂ hence it is positive value so direction is in right side now suppose other case reverse the velocity value the acceleration a = -1.5 m/s₂  which is negative value its mean its acceleration direction is in left side. consider other case figure below.
  v = 4 m/s                                u = 5 m/s
  B←←←←←←←←←←←←A   now suppose body moves point A to B in 1 s then acceleration.

a  = ⎡(-4) - (-5)⎤/1 = 1 which is positive hence direction will be in right side here speed is decreasing from 5 m/s to 4 m/s hence this is called Retardation hence important take sign of acceleration if speed is increasing then acceleration and in the direction of velocity and if speed is decreasing then retardation and direction opposite of velocity. above kinematics equation  v = u + at  this gives the relation between velocity acceleration and time but we want next equation relation with displacement now here initial velocity is u and final velocity is v so what will be average velocity that is 
(u + v)/2 we have study in more details previous previous post about average velocity now we know that displacement = s 
s = average velocity*t = (u + v)/2*t  =( (u +(u + at) )/2*t here we have put the value of v = u + at then simplify equation  we get 
s = ut +1/2at²  this is relation between displacement initial velocity and acceleration suppose if acceleration is not there then put a = 0 equation will become s = ut  which is for uniform motion we have already study now analyse above two equation for third equation.
all parameter involving in accelerated motion are below.
s = displacement,
u = initial velocity,
v = final velocity ,
a = acceleration ,
t = time 
so there are five parameter and so far we have derived two equation.
1 v = u + at    ( here s is missing )
2 s = ut +1/2at²  ( v is missing ) so we have to make one more relation between s and v important acceleration will be not missing in any equation because it is accelerated motion now we have made equation s = (v + u)/2*t
 a = (v - u)/t  now multiply these two equation we will get
as =  (v - u)/t*(v + u)/2*t now simplify equation 
as = (v² - u²)/2 or  v²  = u² + 2as here t is not present .
now one more formula for S𝐧 second suppose body travels distance like in first second 1 m in next second 2 m, in third second 3 m and so on .. hence for nth second t = n
here we have to find the S𝐧 distance in nth second.
use equation s = u +1/2at² suppose in nth second total displacement is S₁ and in (n-1)th second S₂ then we will subtract  S₁ - S₂ will get distance travel in nth second so
S₁ = un +1/2an² put t = n
S₂  = u(n-1) +1/2a(n-1)²  t= (n-1 ) now subtract S₁ - S₂ = S𝐧
S𝐧 =  un +1/2an² - ⎣u(n-1) +1/2a(n-1)² ⎤ now simplify this equation
S𝐧 = u +1/2a(2n-1) this is the distance travel in nth second .
Now important point above all equation  s is displacement not distance v is velocity not speed .
important this equation is only valid for constant acceleration.
thanks for reading 
Know accelerated motion concept
dated 2nd May 2018  

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