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CBSE Class 11-science Answered

obtain equation of Motion for constant acceleration using method of calculus
Asked by yogatamil2005 | 15 Oct, 2020, 03:31: PM
answered-by-expert Expert Answer
Let uniform acceleration be a
 
Acceleration is defined as rate of change of velocity
 
begin mathsize 14px style fraction numerator d v over denominator d t end fraction space equals space a end style  ......................(1)
where v is velocity of an object at time t
 
By integrating above ewqn.(1) , we get
 
begin mathsize 14px style integral subscript u superscript v d v apostrophe space equals space a integral subscript 0 superscript t d t apostrophe space end style
where u is velocity at time t = 0 and v is velocity after time t seconds
 
from above integration we get ,  v - u = a t   or  v = u + ( a t )  ...................... (2)
 
velocity is defined as rate of change of displacement 
 
begin mathsize 14px style v space equals space fraction numerator d S over denominator d t end fraction space equals space u space plus space a space t end style ...................(3)
Where S is distance travelled.
 
By integrating above equation (2) , we get
 
begin mathsize 14px style integral subscript x subscript o end subscript superscript x d S space equals space integral subscript 0 superscript t left parenthesis space u space plus space a space t apostrophe space right parenthesis space d t apostrophe space end style
Where xo is initial position and x is position after time t
 
x - xo = u t + (1/2) a t2 .................. (4)
 
If the object travelled a distance S staring from initial velocity u and attained final velocity v after travelling distance S in time t , then we have
 
S = (1/2) ( u+v ) t  ...............(5)
 
Let us substitute t from eqn.(2) in eqn.(5) as  t = [ ( v - u ) / a ] , then we have
 
S = (1/2) ( u + v ) [ ( v - u ) /a ]    or  v2 = u2 + ( 2 a S )  ..................  (6)
 
Eqn.(2) , eqn.(4) and eqn.(6) are equations of motion
Answered by Thiyagarajan K | 15 Oct, 2020, 04:56: PM
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