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law of conservation of energy in case of freely falling body

Asked by mdowaisraza581 | 16 May, 2023, 10:06: PM

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 E_i 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 E_y 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 K_f is

begin mathsize 14px style K subscript f space equals space 1 half m space v subscript f squared space end style

where v_f is the final speed when body reaches the ground.

Final speed v_f 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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