Question

• Explain the Law of conservation of energy. With each and every derivation.
explain it properly with defination, equation and diagrams.

• No irrelevent answer needed.
• Don't spam. ​

Answer 1

$\huge\tt{\bold{\purple{\underbrace{\red{\mathfrak{Answer♡༄}}}}}}$

$\\ \\ \huge{\fbox{\pink{Defination:-}}} \\ \\ The \: law \: of \: conservation \: of \: energy \: states \: that \\ \: energy \: can \: neither \: be \: created \: nor \: be \: destroyed. \\ \: Although, \: it \: may \: be \: transformed \\ \: from \: one \: form \: to \: another.$

$\huge{\fbox{\pink{Formula:-}}} \\ \\ K_{1} + U_{1} = K_{2} + U_{2} \\ \\ Where, \\ </p><p>K_{1} = Initial \: Kinetic \: Energy \\ </p><p>U_{1} = Initial \: Potential \: Energy \\ </p><p>K_{2} = Final \: Kinetic \: Energy \\ </p><p>U_{2} = Final \: Potential \: Energy$

$\huge{\fbox{\pink{Derivation:-}}}$

Considering the potential energy at the surface of the earth to be zero. Let us see an example of a fruit falling from a tree.

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Consider a point A, which is at height ‘H’ from the ground on the tree, the velocity of the fruit is zero hence potential energy is maximum there.

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$\\ \bold{ E = mgH \longrightarrow1}$

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When the fruit is falling, its potential energy is decreasing and kinetic energy is increasing.

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At point B, which is near the bottom of the tree, the fruit is falling freely under gravity and is at a height X from the ground, and it has speed as it reaches point B. So, at this point, it will have both kinetic and potential energy.

$\\ E = KE + PE \\ \bold{ PE = mg X \longrightarrow2} \\ </p><p>According \: to \: third \: equation \: of \: Motion, \\ {v}^{2} =2g(H - X) \\ \\ \implies \frac{1}{2} m {v}^{2} = \frac{1}{2} m \times 2g(H - X) \\ \\ \implies KE= \frac{1}{2} m \times 2g(H - X) \\ \\ \implies KE = mg(H - X) \\ \\ \\ \bold{ KE = mg (H - X) \longrightarrow3} \\ \\ \\ \\ \\ \\ \green{ Using \: 1, 2, 3} \\ \\ E=mg(H-X)+mgX \\ \implies E= mg(H - X + X) \\ \implies E = mgH$

Similarly, if we see the energy at point C, which is at the bottom of the tree, it will come out to be mgH. We can see as the fruit is falling to the bottom and here, potential energy is getting converted into kinetic energy. So there must be a point where kinetic energy becomes equal to potential energy. Suppose we need to find that height ‘x’ from the ground. We know at that point,

$KE = PE \\ \\ </p><p>\bold{PE = KE = E_{2} \longrightarrow 4} \\ \\ </p><p>E_{2} \: is \: the \: new \: energy. \\ \\</p><p>Where, \: E = mgH_{2} \\</p><p>H_{2}\: is \: the \: new \: height.\\</p><p>As,\: the\: body\: is\: at\: height\: x\: from\: the\: ground, \\ \\ </p><p>\bold{PE = mgX \longrightarrow5}\\ \\ \\</p><p>\green{Using\: 4\: and\: 5\: we\: get,} \\ \\ mgX = \frac{mgH}{2} </p><p> \\ \\ \implies X = \frac{H}{2}$

$\\ \\ \\ \\ \\ \huge{note:}$

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Answer 2

Law of Conservation of Energy :-

The law of conservation of energy states that energy can neither be created nor be destroyed. Although, it may be transformed from one form to another. If you take all forms of energy into account, the total energy of an isolated system always remains constant. All the forms of energy follow the law of conservation of energy. In brief, the law of conservation of energy states that

• In a closed system, i.e., a system that is isolated from its surroundings, the total energy of the system is conserved.

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