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law of conservation of energy in case of freely falling body
Asked by mdowaisraza581 | 16 May, 2023, 22:06: PM
answered-by-expert Expert Answer
Let the body is dropped from height h . Just before dropping it has only potential energy (mgh) ,
where m is mass of body, g is acceleratio due to gravity and h is height.
 
Hence intial total energy Ei  is
 
 begin mathsize 14px style E subscript i space equals space m space g space h end style  ................................... (1) 
 
Let us consider it has fallen a distance y from initial point .
 
Potential energy U at this point is 
 
begin mathsize 14px style U space equals space m space g space left parenthesis h minus y right parenthesis end style   ...............................(2)
 
Speed of body at this point is determined from the following equation
 
begin mathsize 14px style v squared space equals space 2 g y end style
Hence kineti energy K at this point is 
 
begin mathsize 14px style K space equals space 1 half m space v squared space space equals space 1 half m space left parenthesis space 2 space g space y space right parenthesis space equals space left parenthesis space m space g space y space right parenthesis space end style
Total energy Ey at this point is sum of potential energy and kinetic energy
 
begin mathsize 14px style E subscript y space equals space U space plus space K space equals space m g left parenthesis h minus y right parenthesis space plus space left parenthesis space m g y space right parenthesis space equals space m g h end style  ...........................(3)
 
Finally, when the body reaches ground , its potential energy is zero and has only kinetic energy.
 
Final kinetic energy Kf is
 
begin mathsize 14px style K subscript f space equals space 1 half m space v subscript f squared space end style
where vf is the final speed when body reaches the ground. 
 
Final speed vf is determined from the following equation
 
begin mathsize 14px style v subscript f superscript 2 space equals space 2 space g space h space end style
Hence final kinetic energy , or total energy at ground is
 
 begin mathsize 14px style E subscript f space equals space K subscript f space equals space 1 half m space v subscript f superscript 2 space equals space 1 half m space left parenthesis space 2 space g space h space right parenthesis space equals space m space g space h end style  ......................................(4)
 
Hence from eqn.(1) , eqn.(3) and eqn.(4) , we see that total energy always remain same as ( m g h ) . 
 
This is due to conservation of energy.  At a height h, all total energy is in the form of potential energy.
 
After falling a distance y from starting point , potential energy decreases but kinetic energy increases .
 
We found that sum of potential energy and kinetic energy is ( m g h ) .
 
When the body is reaching ground, just before hitting the ground , it has only kinetic energy that equals ( m g h ) .
 
Hence loss in potential energy equals gain in kinetic energy so that sum of kinetic enrgy and potential energy at all the points is same.
Answered by Thiyagarajan K | 17 May, 2023, 11:27: AM
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