Question

show that the total mechanical energy of a body falling freely under gravity is conserved.

Answer 1

Hey!

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Let us consider a body of mass m placed at A

h = AB = Height of body from ground
s = Distance if any point C from A
g = Acceleration due to gravity
v1 = Velocity of the body at C
v = Velocity of body at B

Velocity at point A = 0

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(i) Mechanical energy at Point A

Potential energy at A = E (pA) = mgh
Kinetic energy at A = E (kA) = 0
Total mechanical energy at A (Ea) = E (pa) + E (kA)

= mgh + 0 = mgh -------- (i)

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(ii) At point C

When body moves from A to C, it covers distance 's'. and v1 us velocity at C then from,

v^2 - u^2 = 2as
v1^2 - 0 = 2 gs

Kinetic energy at C = E (kC) = 1/2 mv^2
= 1/2 m (2gs)
= mgs

Potential energy at C = E (pC) = mg (h - s)

Total mechanical energy at C = E (C) =

E (C) = E (pC) + E (kC) = mg (h-s) + mgs = mgh ------ (ii)

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(iii) At point B

v2 = 2gh

Kinetic energy at B = E (kB) = 1/2 mv^2
= 1/2 m (2gh) = mgh

Potential energy at B = E (pB) = 0

Total mechanical energy = E (B) =

E (B) = E (kB) + E (pB) = mgh + 0 = mgh ----- (iii)

From (i) , (ii) and (iii)

E (A) = E (B) = E (C)

mgh = mgh = mgh

Total mechanical energy was same at all the three points. This proves that -

The total mechanical energy of the body throughout the free fall is conserved.

NOTE - We have neglected effect of AIR RESISTANCE on the motion of body.

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Hope it helps...!!!

Answer 2

Hey there

Let's explore something new before proofing this statement , Why I'm telling you ? It is because the theories I will discuss which will be use over here so it will be better to you to understand it First .

• What is mechanical Energy ?

→ Mechanical energy is the sum of the kinetic and potential Energy

Thus if M is the machanical energy then it can be written as

→ M = KE + PE ..... ( 1 )

• What do you mean by the Conservation of energy ?

→ This statement states that , Energy can neither created nor destroyed but it can be transferred from one form to another

For example :-

• Have you ever heard about photosynthesis ? this is the real life example of the energy conversion , light energy is converted to Chemical energy

•What did you eat today ? You know what the food that you eat is digested and give energies to you by the process respiration

well we have learnt so much thing yet

Now let's proof this statement

Suppose you are climbing up to Mount everest how good it is xD ,

then we know a body have mass , this body is affected by the gravity , and the body goes to the height , due to change in the position up it produces velocity

these can be denoted as

Velocity , = V

Mass , = m

Gravity ,/ acceleration due to free fall = g

height , = h

[ see the attached picture no -1 ]
where

$AT \: POINT \: - A$

Kinetic energy , KE = 0 [ since no any motion at that time ]

Potential Energy = mgh

From equation [ 1 ]

Mechanical Energy = 0 + mgh= mgh

$AT \: POINT \: - B$

Suppose a body with m is falling from the hight ( h - x ) with velocity vB at point B it produces gravity g covered the distance x

then

•Potential Energy , PE = mg ( h - x )

By 3rd equation of motion:-

${v}^{2} = {u}^{2} + 2as \\ or \\ \\ { vb\: }^{2} = 0 + 2gs \\ {Vb \: }^{2} = 2gx$

• kinetic energy

$= \frac{1}{2} m {Vb \: }^{2} \\ \\ = \frac{1}{2} m.2gx \\ \\ = mgx \\ \\ mechanical \: energy \: = mg \: (h - x) + mgx \: \\ = mgh \\ \\ AT$
AT POINT C :-

where

PE =0

Velocity , Vc

PE = 0

KE = mgh

Hence mechanical energy = mgh

we
see that at Different point the value of mechanical energy is the same Hence it is proved that it is conserved at different points

Hope this helps you ☺