Peoples Physics Book Version-2

http://www.ck12.org Chapter 9. Energy and Force Version 2

9.3 Work-Energy Principle

The reason the concept of work is so useful is because of a theorem, called thework-energy principle, which states
thatthe change in an object’s kinetic energy is equal to the net work done on it:

∆Ke=Wnet[2]

Although we cannot derive this principle in general, we can do it for the case that interests us most: constant
acceleration. In the following derivation, we assume that the force is along motion. This doesn’t reduce the generality
of the result, but makes the derivation more tractable because we don’t need to worry about vectors or angles.

Recall that an object’s kinetic energy is given by the formula:

Ke=^12 mv^2 [3]

Consider an object of massmaccelerated from a velocityvitovf under a constant force. The change in kinetic
energy, according to [2], is equal to:

∆Ke=Kei−Ke f=^12 mv^2 f−^12 mv^2 i=^12 m(v^2 f−v^2 i)[4]

Now let’s see how much work this took. To find this, we need to find the distance such an object will travel under
these conditions. We can do this by using the third of our ’Big three’ equations, namely:

vf^2 =vi^2 + 2 a∆x[5]

alternatively,

∆x=

vf^2 −vi^2

2 a

[6]

Plugging in [6] and Newton’s Third Law,F=ma, into [2], we find:

W=F∆x=ma×

vf^2 −vi^2

2 a

=^12 m(v^2 f−v^2 i)[7],

which was our result in [4].

Using the Work-Energy Principle

The Work-Energy Principle can be used to derive a variety of useful results. Consider, for instance, an object dropped
a height∆hunder the influence of gravity. This object will experience constant acceleration. Therefore, we can again
use equation [6], substituting gravity for acceleration and∆hfor distance:

∆h=

vf^2 −vi^2

2 g

multiplying both sides by $mg$, we find:

mg∆h=m
g

vf^2 −vi^2

(^2)
g
=∆Ke[8]
In other words, the work performed on the object by gravity in this case ismg∆h. We refer to this quantity as
gravitational potential energy; here, we have derived it as a function of height. For most forces (exceptions are
friction, air resistance, and other forces that convert energy into heat), potential energy can be understood as the
ability to perform work.