CBSE Class 11-science Answered
obtain equation of Motion for constant acceleration using method of calculus
Asked by yogatamil2005 | 15 Oct, 2020, 15:31: PM
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](https://images.topperlearning.com/topper/tinymce/cache/4bbf14371183140ae295ab4b9dfe4198.png)
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](https://images.topperlearning.com/topper/tinymce/cache/69c68d21e7885f4d2ecc50aa13bb1f25.png)
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](https://images.topperlearning.com/topper/tinymce/cache/b5c08fd95f103327a22b4a5fca516ece.png)
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](https://images.topperlearning.com/topper/tinymce/cache/e273b9578d13e427871004d676b2ea83.png)
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, 16:56: PM
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