Conservation of mechanical energy

Mechanical energy

Mechanical energy is the sum of kinetic energy and potential energy:

$E_{mech}=K+U$

where

K=rac{1}{2}mv^2

and $U$ may include gravitational, spring, or other conservative potential energies.

When only conservative forces do work, mechanical energy is conserved:

$K_i+U_i=K_f+U_f$

This principle often solves problems more simply than Newton's second law.

Why mechanical energy is conserved

From the work-energy theorem,

$W_{net}=Delta K$

If the only work is done by conservative forces, then

$W_{conservative}=-Delta U$

Therefore

$Delta K=-Delta U$

or

$Delta K+Delta U=0$

This means

$Delta(K+U)=0$

so total mechanical energy remains constant.

Falling object example

Consider an object dropped from height $h$ with negligible air resistance. Choose $U=0$ at the ground. Initially,

$K_i=0$

$U_i=mgh$

Just before hitting the ground,

$U_f=0$

and

K_f=rac{1}{2}mv^2

Conservation gives

mgh=rac{1}{2}mv^2

so

$v=sqrt{2gh}$

The mass cancels, showing that all objects fall with the same speed from the same height in this ideal model.

Spring example

For a mass attached to an ideal spring on a frictionless surface,

E=rac{1}{2}mv^2+rac{1}{2}kx^2

At maximum displacement $A$, speed is zero, so

E=rac{1}{2}kA^2

At equilibrium $x=0$, spring potential energy is zero and kinetic energy is maximum:

rac{1}{2}mv_{max}^2=rac{1}{2}kA^2

Thus

v_{max}=Asqrt{rac{k}{m}}

Including non-conservative work

If non-conservative forces such as friction do work, mechanical energy is not conserved. The energy equation becomes

$K_i+U_i+W_{nc}=K_f+U_f$

where $W_{nc}$ is work by non-conservative forces.

For kinetic friction over distance $d$,

$W_{friction}=-f_kd$

The lost mechanical energy becomes thermal energy and other microscopic forms.

Choosing the system

Energy conservation depends on the system chosen. If you include Earth and the falling object as one system, gravitational potential energy belongs to the system. If you analyze only the object, gravity is an external force doing work.

Choosing the system clearly helps avoid confusion about whether energy is transferred into, out of, or within the system.

When energy methods are best

Energy methods are especially useful when time is not needed. If the question asks for speed after falling a certain height, compression of a spring, or distance traveled before stopping, energy may be faster than force analysis.

However, energy alone may not provide direction, time, or detailed acceleration. It is one tool among several.

The big idea

Mechanical energy is conserved when only conservative forces do work. This allows motion problems to be solved by comparing initial and final energy rather than analyzing every moment. When friction or other non-conservative forces act, their work accounts for changes in mechanical energy.