KAHIBARO
Table of Contents
Core Idea
Conservation of mechanical energy is the rule that the total of kinetic energy and potential energy stays constant when only conservative forces act on a system.
Mechanical energy is defined as
$$
E_{\text{mech}} = K + U
$$
where $K$ is kinetic energy and $U$ is potential energy.
If no non conservative forces, such as friction or air resistance, change the energy, then mechanical energy is conserved:
$$
K_i + U_i = K_f + U_f
$$
This means energy can change form, for example from potential energy into kinetic energy, but the total mechanical energy remains the same.
For conservation of mechanical energy to apply in its simplest form, the system must be affected only by conservative forces.
The key equation is
$$
K_i + U_i = K_f + U_f
$$
or equivalently
$$
\Delta K + \Delta U = 0
$$
What the Equation Means
Suppose an object falls downward. As it loses gravitational potential energy, it gains kinetic energy. The decrease in $U$ is exactly balanced by the increase in $K$.
If a spring is compressed and then released, elastic potential energy changes into kinetic energy. Again, the total mechanical energy stays the same if no energy is lost to friction or other non conservative effects.
This idea is powerful because it often lets us solve motion problems without finding the force and acceleration at every moment.
Standard Mathematical Form
For two positions, labeled 1 and 2, conservation of mechanical energy is usually written as
$$
K_1 + U_1 = K_2 + U_2
$$
Using common forms of energy, this can become:
$$
\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2
$$
for motion under gravity near Earth's surface, or
$$
\frac{1}{2}mv_1^2 + \frac{1}{2}kx_1^2 = \frac{1}{2}mv_2^2 + \frac{1}{2}kx_2^2
$$
for a spring system.
Sometimes both kinds of potential energy appear in the same problem.
Interpreting Energy Changes
A useful way to think about the rule is that one part goes up while another goes down.
| Situation | Kinetic energy | Potential energy | Total mechanical energy |
|---|---|---|---|
| Object falling | Increases | Decreases | Constant |
| Object rising | Decreases | Increases | Constant |
| Spring releasing | Increases | Decreases | Constant |
| Spring compressing | Decreases | Increases | Constant |
The total does not have to be split equally between kinetic and potential energy. It is only the sum that must remain constant.
Example, Falling Object
Consider a mass $m$ dropped from height $h$ with initial speed zero. At the top,
$$
K_i = 0, \qquad U_i = mgh
$$
At a lower point where the height is zero,
$$
K_f = \frac{1}{2}mv^2, \qquad U_f = 0
$$
Conservation of mechanical energy gives
$$
mgh = \frac{1}{2}mv^2
$$
so
$$
v = \sqrt{2gh}
$$
The mass cancels, so the result does not depend on the object's mass.
Example, Vertical Throw
An object is thrown upward with speed $v_0$. At the highest point, its speed becomes zero. Let the launch point have height zero. Then
$$
\frac{1}{2}mv_0^2 = mgh_{\max}
$$
which gives
$$
h_{\max} = \frac{v_0^2}{2g}
$$
This shows that the initial kinetic energy is fully converted into gravitational potential energy at the top.
Example, Mass on a Spring
A block attached to a spring moves on a frictionless surface. At maximum compression or extension, the speed is zero, so all the mechanical energy is elastic potential energy:
$$
E_{\text{mech}} = \frac{1}{2}kx^2
$$
At the equilibrium position, the spring potential energy is zero, so all the mechanical energy is kinetic:
$$
E_{\text{mech}} = \frac{1}{2}mv^2
$$
Thus,
$$
\frac{1}{2}kx^2 = \frac{1}{2}mv^2
$$
for those two special positions.
Choosing a Reference Level
Potential energy depends on the choice of zero level, especially gravitational potential energy near Earth. You may choose any convenient reference height for $U = 0$.
Different choices of zero do not change the physics, because only differences in potential energy matter.
You may choose the zero of potential energy wherever convenient, but you must use that choice consistently throughout the problem.
When This Rule Can Be Used
Conservation of mechanical energy is most useful when you compare two positions and do not need details of the motion in between.
Typical cases include motion under gravity alone, spring motion without friction, and combinations of conservative forces.
If friction, drag, or other non conservative forces do important work, then mechanical energy is not conserved by itself. In that case, the broader conservation of total energy still holds, but some mechanical energy is transformed into thermal energy or other forms.
Visual Picture
This drawing shows a typical situation where one form of energy decreases while the other increases, but their sum stays fixed.
Problem Solving Strategy
To use conservation of mechanical energy, identify the initial and final states, write the kinetic and potential energies at each state, and set the totals equal.
A compact structure is:
$$
K_i + U_i = K_f + U_f
$$
Then substitute the correct expressions, such as $\frac{1}{2}mv^2$, $mgh$, or $\frac{1}{2}kx^2$.
Do not assume mechanical energy is conserved unless the forces involved are conservative, or unless non conservative effects are negligible.
Final Summary
Conservation of mechanical energy states that in a system acted on only by conservative forces, the sum of kinetic and potential energy remains constant:
$$
E_{\text{mech}} = K + U = \text{constant}
$$
This leads to
$$
K_i + U_i = K_f + U_f
$$
It is one of the most useful tools in mechanics because it connects speed, height, and position directly, without needing the full details of the motion.