A Quick Note on Work-Energy Theorem Derivation

The work-energy theorem derived understands energy in relation to force. In the work-energy theorem formula, the role of displacement is also crucial. The work-energy theorem is derived from several other aspects. These will be elaborated on in the following sections. However, knowing what energy is in physics is essential to understanding the work-energy theorem derivation. 

What is energy?

Energy is defined as the capacity to do work. Modern civilization is viable because human beings have found a way to shift energy from one state to another and use it to do work. 

Energy is classified into two main types:

• Potential or Stored Energy
• Kinetic or Working Energy

Potential energy is the energy that is conserved in a stationary object. Whereas kinetic energy is the energy that is produced when the object is in motion. 

Energy is directly related to motion or movement. It is produced when the two objects or an object are in some kind of motion. For instance, cooking involves various movements such as chopping, stirring, straining, etc. In each of these processes, there are different levels of energy involved and are produced.

The law of thermodynamics states that energy can never be created or destroyed. It can only be transmitted from one form to the other. For instance, when you push a box, it changes its form of energy. It has potential energy when it is at rest. However, it changes its form into kinetic energy once it is pushed. 

Energy is broadly classified into various types, and one such part is work energy. 

What Is Work-Energy?

According to the work-energy concept, a significant amount of work is done to produce energy. It believes that a body exhibits kinetic energy when a significant amount of work is done by the resulting force on the body. As per the statement, the work-energy concept states “an increase in the kinetic energy” in a rigid body when an “equal amount of positive work” is applied to the body, with the resultant force in action. 

Work-energy is calculated in terms of Joules or (J). This concept also explains the reverse situation. That is, if there is no force acting on the body, then the body exhibits potential energy, and there is no force involved. Kinetic energy will only increase when the work done on the body is positive. That is, the force applied pushed the body towards kinetic energy. 

In the work-energy theorem derived, displacement also plays an essential role. Here, displacement is defined as the direction in which the body moves when it has kinetic energy. It also measures how far the body travels and what could have been the intensity of the kinetic energy. 

Some common examples of work-energy are:

• Lifting weight from the ground
• Running up and down a slope
• The work of a hydraulic arm
• Starting a the car to drive

The work-energy can be calculated as the work-energy theorem. The work-energy theorem derived includes various components that are elaborated in the next section. 

Work-Energy Theorem Derivation

The work ‘W’ done by the net force on a particle is the same as the change in the object’s kinetic energy (KE). 

Let us bear in mind a case where the resulting pressure ‘F’ is consistent in each course and value. It is parallel to the speed of the object. The object is shifting with constant acceleration alongside a direct line. The relation between the acceleration and the total pressure is given via the equation “F = ma” (Newton’s 2nd law of motion), and the particle’s displacement ‘d’ may be decided from the equation:

Vf2=Vi2+ 2ad

Where

Vf = final velocity

Vi = initial velocity

a = acceleration 

d = displacement

By this, we obtain, 

W=ΔK.E.=12mVf2–12mVi2

Here, the work done on the total force is further calculated as its magnitude’s product, that is, F=ma. This also includes the object’s displacement. 

W=Fd=maVf2–Vi22a=12mVf2–12mVi2=K.Ef–K.Ei=ΔK.E.

This is the final equation as the work-energy theorem derived from the previous derivations. 

To obtain this derivation of the work energy, the initial steps also include the derivation of kinetic energy. The derivation of kinetic energy works as the principle in determining the intensity and the displacement of the object that would travel. It further elaborates on the kinetic energy produced when the resulting force acts on the object.

Conclusion 

The above information broadly elaborates on what energy is. It further discusses the types of energy and what work energy is. Work-energy theorem refers to the calculation of energy produced when a work is in action. There are complete mathematical steps on how the work-energy theorem is derived. The importance of the derivation of kinetic energy in the work-energy theorem is also highlighted.