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

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 x_o is initial position and x is position after time t

x - x_o = 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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