Question

what is the derivation of potential energy and Kinetic energy​

Answer 1

Answer:

The kinetic energy of a body is the energy that it possessed due to its motion. Kinetic energy can be defined as the work needed to accelerate an object of a given mass from rest to its stated velocity. The derivation of kinetic energy is one of the most common questions asked in the examination. Students must understand the kinetic energy derivation method properly to excel in their examination.

Kinetic energy depends upon the velocity and the mass of the body. If the velocity of the body is zero, then the kinetic energy will also be zero. The derivation of kinetic energy is given below so that students can understand the concept more effectively. the kinetic energy formula derivation can be done using algebra and calculus. Both the methods are explained below.

Derivation of Kinetic Energy using Algebra

The kinetic energy derivation using only algebra is one of the best ways to understand the formula in-depth.

Starting with the work-energy theorem and then adding Newton’s second law of motion we can say that,

Derivation Of Kinetic Energy

Now, taking the kinematics equation and rearranging it, we get

Derivation Of Kinetic Energy

Combining the 2 expressions we get,

Derivation Of Kinetic Energy

Derivation Of Kinetic Energy

Now we already know that kinetic energy is the energy that it possessed due to its motion. So the kinetic energy at rest should be zero. Therefore we can say that kinetic energy is:

Derivation Of Kinetic Energy

Derivation of Kinetic Energy using Calculus

The derivation of kinetic energy using calculus is given below. To derive an expression for kinetic energy using calculus, we will not need to assume anything about the acceleration.

Starting with the work-energy theorem and Newton’s second law of motion we can say that

Derivation Of Kinetic Energy

Now rearranging the differential terms to get the function and the integral into an agreement.

Derivation Of Kinetic Energy

Derivation Of Kinetic Energy

Now, we know that the kinetic energy of a body at rest is zero. So we can say that the kinetic energy is:

Derivation Of Kinetic Energy

Explanation:

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Answer 2

$\mathfrak{ Derivation \: of \: Potential \: Energy}$

$\mathsf{Things \: to \: know \: before\: Derivation}$

$\mathsf{\implies Work \: done = Fs = F × height = \: Fh}$

$\mathsf{\implies Work = Energy}$

$\mathsf{\implies Force = mass \: × \: acceleration \: = ma}$

$\sf{\implies Acceleration \:due \:to \:gravity =\: 'g'}$

$\mathsf{\implies Potential \: Energy \: is \: also \: written \: as \: E_p}$

$\mathsf{Derivation}$

$\mathsf{\hookrightarrow E_p = Work done = Fs = Fh }$

$\mathsf{\hookrightarrow E_p \: = \: Fh \: = ma × h }$

$\mathsf{\hookrightarrow E_p\: = \:Fh \: = mg × h }$

$\mathsf{\hookrightarrow E_p\: = \:mgh }$

$\mathfrak{Derivation \: of \:Kinetic \: Energy}$

$\mathsf{Things \: to \: know \: before\: Derivation}$

$\mathsf{\implies Work done = Fs}$

$\mathsf{\implies v² = u² + 2as}$

$\mathsf{\implies s = \frac{v² - u²}{2a}}$

$\mathsf{\implies u = 0 m/s \: for \:a \: body \: starting \: from \: rest}$

$\mathsf{\implies Work \: Done = \: Energy}$

$\mathsf{\implies Kinetic \: Energy \: is \: also \: written \: as \: E_k}$

$\mathsf{\implies Force = mass \: × \: acceleration \: = ma}$

$\mathsf{Derivation}$

$\mathsf{\hookrightarrow E_k = Work done = Fs }$

$\mathsf{\hookrightarrow \: = \: Fs \: = ma × s }$

$\mathsf{\hookrightarrow E_k = m × \frac{v² - u²}{2a} × a}$

$\mathsf{\hookrightarrow E_k = \frac{1}{2}mv²}$