1. Definition

The Principle of Conservation of Mechanical Energy states:

If only conservative forces act on a system, the total mechanical energy of the system remains constant.

Mechanical energy is the sum of:

• kinetic energy
• potential energy

\(E_{mech}=K+U\)

2. Conservation Equation

When no non-conservative forces do work:

\(K_i+U_i=K_f+U_f\)​

where:

• \(K_i\) = initial kinetic energy
• \(U_i\) = initial potential energy
• \(K_f\)​ = final kinetic energy
• \(U_f\) = final potential energy

3. Why Mechanical Energy Is Conserved

For conservative forces:

\(W_c=-\Delta U\)

From the Work–Energy Theorem:

\(W_{net}=\Delta K\)

If only conservative forces act:

\(W_c=\Delta K\)

Substituting:

\(-\Delta U=\Delta K\)

Rearranging:

\(\Delta K+\Delta U=0\)

Therefore:

\(K+U=constant\)

4. Mechanical Energy

Mechanical energy consists of:

Kinetic Energy

Gravitational Potential Energy

\(U_g=mgh\)

Elastic Potential Energy

\(U_s=\frac{1}{2}kx^2\)

5. Falling Object Example

Consider a ball dropped from height h.

Initially:

\(K_i=0\)

\(U_i=mgh\)

As the ball falls:

• gravitational potential energy decreases
• kinetic energy increases

Just before reaching the ground:

\(U_f=0\)

\(K_f=mgh\)

Mechanical energy remains constant throughout the motion.

6. Object Thrown Upward

As the object rises:

\(K \downarrow\)

\(U_g \uparrow\)

At the highest point:

\(v=0\)

Therefore:

\(K=0\)

Potential energy reaches its maximum value.

7. Spring-Mass System

A compressed spring stores:

\(U_s=\frac{1}{2}kx^2\)

When released:

• spring potential energy decreases
• kinetic energy increases

At equilibrium:

• spring potential energy is minimum
• kinetic energy is maximum

8. Non-Conservative Forces

Mechanical energy is not conserved when non-conservative forces perform work.

Examples:

• friction
• air resistance

In these situations:

\(W_{nc}=\Delta E_{mech}\)​

or

\(W_{nc}=(K_f+U_f)-(K_i+U_i)\)

Mechanical energy changes because some energy is transformed into:

• thermal energy
• sound
• deformation

9. Advantages of the Energy Method

Using conservation of energy often avoids solving for:

• acceleration
• forces
• time

This makes many AP Physics C problems significantly easier than applying Newton’s Second Law directly.

10. Problem-Solving Strategy

Step 1

Identify initial and final states.

Step 2

Determine all relevant forms of energy.

• kinetic
• gravitational potential
• elastic potential

Step 3

Check whether non-conservative forces are present.

Step 4

Apply either:

\(K_i+U_i=K_f+U_f\)​

or

\(K_i+U_i+W_{nc}=K_f+U_f\)​​

11. Physical Interpretation

Conservation of mechanical energy does not mean energy disappears or appears.

Instead, energy continuously changes form:

\(U \leftrightarrow K\)

while the total mechanical energy remains constant when only conservative forces act.

Summary

Mechanical energy is:

\(E_{mech}=K+U\)

When only conservative forces act:

\(K_i+U_i=K_f+U_f\)​

Key ideas:

• energy can change form
• kinetic and potential energy exchange continuously
• total mechanical energy remains constant
• non-conservative forces cause mechanical energy to change

Conservation of Mechanical Energy is one of the most powerful tools in AP Physics C Mechanics because it allows complex motion problems to be solved without directly analyzing forces and acceleration.