Answer:
Step-by-step explanation:
The law of conservation of energy states that energy can neither be created nor destroyed, but it can be converted from one form to another. In the case of an object in motion, its energy can be transformed between different types, such as kinetic energy (K.E.) and work done.
To understand the relationship between final kinetic energy, K.E., and total work done, let's consider an object of mass m that initially has a kinetic energy of K.E.i. As work is done on the object, its kinetic energy changes. The work done on the object can be expressed as the product of the force applied to the object and the distance over which the force is applied, given by W = Fd.
Using the work-energy theorem, we can state that the work done on an object is equal to the change in its kinetic energy. Therefore, the work done on the object is W = K.E.f - K.E.i, where K.E.f represents the final kinetic energy of the object.
By rearranging the equation, we find that K.E.f = W + K.E.i. This equation shows that the final kinetic energy of the object is equal to the sum of the work done on the object and its initial kinetic energy.
Now let's prove this relationship mathematically. We know that work done is equal to the integral of force with respect to displacement: W = ∫F dx. Using Newton's second law, F = ma, we can rewrite the equation as W = ∫m a dx.
By using the product rule of differentiation (∫u dv = uv - ∫v du), we can rewrite the equation as W = ∫m (dv/dt) dx. Applying the chain rule (∫u(dv/dx) dx = uv - ∫v(du/dx) dx), we have W = ∫(m dx/dt) dv.
Since dx/dt is nothing but velocity (v), we can simplify the equation as W = ∫m dv. Introducing limits of velocities from initial (vi) to final (vf), we have W = ∫(vi to vf) m dv.
Integrating the equation gives W = (1/2)mvf^2 - (1/2)mvi^2. This equation represents the work done on the object.
Combining this result with the initial equation K.E.f = W + K.E.i, we have K.E.f = (1/2)mvf^2 - (1/2)mvi^2 + K.E.i.
Simplifying further, K.E.f = (1/2)m(vf^2 - vi^2) + K.E.i.
Recognizing that (vf^2 - vi^2) can be rewritten as (vf + vi)(vf - vi), we have K.E.f = (1/2)m(vf + vi)(vf - vi) + K.E.i.
Since (vf + vi) is the sum of final and initial velocities, we can express it as 2v. Substituting this in the equation, we get K.E.f = mv(vf - vi) + K.E.i.
Finally, recognizing that mv(vf - vi) is nothing but the change in momentum, Δp = mv(vf - vi), we have K.E.f = Δp + K.E.i.
This equation states that the final kinetic energy of an object is equal to the change in its momentum plus the initial kinetic energy. Thus, it proves that the final kinetic energy is indeed equal to the sum of the work done on the object and its initial kinetic energy.