Conservation of Energy

Introduction

students, imagine a roller coaster starting high on a hill and speeding down the track 🎢. The coaster does not need to “create” motion from nowhere. Instead, energy changes form as the coaster moves. In this lesson, you will learn the main ideas behind conservation of energy, how to use it in problem solving, and how it connects to work, kinetic energy, potential energy, and power.

Learning objectives

Explain the meaning of conservation of energy and the related vocabulary.
Apply conservation of energy to motion problems in AP Physics C: Mechanics.
Connect conservation of energy to work, energy, and power.
Summarize why conservation of energy is a central idea in mechanics.
Use examples and evidence to reason about energy changes in physical systems.

The big idea is simple: for a closed system, energy is not created or destroyed. It can only be transferred or transformed. In mechanics, this often means that energy moves between kinetic energy and potential energy. When forces like friction are present, some mechanical energy becomes thermal energy, but the total energy of the full system still stays constant.

What Conservation of Energy Means

The law of conservation of energy says that the total energy of an isolated system remains constant. In physics, “isolated” means no energy enters or leaves the system. For many AP Physics C problems, this idea is used with mechanical energy, which is the sum of kinetic and potential energy.

The kinetic energy of an object is given by $K = \frac{1}{2}mv^2$ where $m$ is mass and $v$ is speed. Gravitational potential energy near Earth is usually written as $U_g = mgh$ where $g$ is the acceleration due to gravity and $h$ is height relative to a chosen reference level. Spring potential energy is $U_s = \frac{1}{2}kx^2$ where $k$ is the spring constant and $x$ is the stretch or compression.

For a system with only conservative forces acting, mechanical energy is conserved:

$$K_i + U_i = K_f + U_f$$

This equation means the total of kinetic and potential energy at the start equals the total at the end. The subscript $i$ means initial, and the subscript $f$ means final.

A conservative force is a force for which the work done does not depend on the path taken, only on the starting and ending positions. Gravity and spring force are conservative. Friction is not conservative because it depends on the path length and transforms mechanical energy into thermal energy.

Connecting Energy Changes to Motion

Conservation of energy is powerful because it lets you relate motion at different points without tracking every detail of the path. This is especially useful when an object moves through changing heights or through a spring system.

Suppose a ball is dropped from rest from height $h$. At the top, its energy is mostly gravitational potential energy. As it falls, $U_g$ decreases while $K$ increases. Just before hitting the ground, if air resistance is ignored, almost all of the original potential energy has become kinetic energy.

Using conservation of energy:

$$mgh_i + \frac{1}{2}mv_i^2 = mgh_f + \frac{1}{2}mv_f^2$$

If the object starts from rest, then $v_i = 0$. If the final height is chosen as $h_f = 0$, then the result becomes:

$$mgh = \frac{1}{2}mv^2$$

Solving for speed gives:

$$v = \sqrt{2gh}$$

This is a common AP Physics C result for free fall without air resistance. Notice that the mass $m$ cancels, which means objects dropped from the same height reach the same speed, assuming only gravity acts.

Example 1: Sliding Down a Ramp

students, picture a block sliding down a frictionless ramp from a height of $h$. The block starts from rest. Because the ramp is frictionless, only gravity does work, and mechanical energy is conserved.

At the top:

$$K_i = 0, \quad U_i = mgh$$

At the bottom:

$$U_f = 0, \quad K_f = \frac{1}{2}mv^2$$

Set initial and final energies equal:

$$mgh = \frac{1}{2}mv^2$$

Cancel $m$ and solve:

$$v = \sqrt{2gh}$$

This result tells you something important: the final speed depends on the vertical drop, not on the steepness of the ramp. A long gentle ramp and a short steep ramp can give the same speed at the bottom if the height drop is the same and friction is absent. 🛷

If friction were present, then some energy would be converted into thermal energy. The final speed would be smaller than the frictionless prediction.

Example 2: Mass on a Spring

A spring is another classic conservation of energy situation. Imagine compressing a spring and launching a cart across a frictionless track.

If the spring starts compressed by $x$ and the cart begins at rest, then the initial energy is spring potential energy:

$$U_{s,i} = \frac{1}{2}kx^2$$

When the spring returns to its natural length, the spring potential energy becomes zero and the cart moves with speed $v$:

$$K_f = \frac{1}{2}mv^2$$

So,

$$\frac{1}{2}kx^2 = \frac{1}{2}mv^2$$

Solving for $v$ gives:

$$v = x\sqrt{\frac{k}{m}}$$

This equation shows that a stiffer spring or larger compression produces a greater speed, while a larger mass lowers the speed. This kind of problem is common because it links force, energy, and motion in one clean model.

Conservative Forces, Nonconservative Forces, and Work

In AP Physics C, energy methods are closely tied to the work-energy theorem. The work done by the net force changes kinetic energy:

$$W_{\text{net}} = \Delta K$$

If only conservative forces act, then changes in potential energy account for the work done by those forces. For gravity or springs, we can use potential energy functions instead of calculating work along the path.

When nonconservative forces like friction act, mechanical energy is not conserved. Instead, the relationship becomes:

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

Here, $W_{\text{nc}}$ is the work done by nonconservative forces. If friction acts opposite the motion, then $W_{\text{nc}}$ is negative. That means mechanical energy decreases, even though total energy remains conserved when thermal energy is included.

For example, if a sled slides across snow, friction converts some of the sled’s mechanical energy into thermal energy in the sled and snow. The sled slows down because less mechanical energy remains available for motion.

Choosing the Right System and Reference Level

A major skill in conservation of energy problems is defining the system carefully. The system might include the object, Earth, a spring, or all of these together. Choosing a system helps you know which energies belong in the equation.

You also need a reference level for gravitational potential energy. Since only changes in gravitational potential energy matter, you can choose $h = 0$ wherever is convenient. For a problem with a hill, the bottom of the hill is often chosen as the zero level. For a drop from a cliff, the ground may be a natural choice.

What matters is consistency. If you set one point to zero, use that same reference level everywhere in the problem. The final answer for speed or height will not depend on where you placed the zero of potential energy, because only differences in $U_g$ affect the result.

Example 3: Using Energy with a Pendulum

A pendulum is a great example of energy changing back and forth. At the highest point of its swing, the pendulum bob has the greatest gravitational potential energy and the least kinetic energy. At the lowest point, it has the greatest kinetic energy.

If a bob drops a vertical distance $h$ from its starting point to the lowest point, then conservation of energy gives:

$$mgh = \frac{1}{2}mv^2$$

So,

$$v = \sqrt{2gh}$$

This speed is the same result you found in the ramp example because only the vertical change in height matters. In a real pendulum, some energy is lost to air resistance and friction at the pivot, so the motion slowly dies out. That is evidence that mechanical energy is not perfectly conserved in the presence of nonconservative forces.

How to Solve Conservation of Energy Problems

A reliable strategy is:

Identify the system.
Decide whether mechanical energy is conserved.
Write initial and final energy expressions.
Include all relevant forms of energy, such as $K$, $U_g$, and $U_s$.
Add nonconservative work if friction or another nonconservative force is present.
Solve algebraically before substituting numbers.

This method is often faster than using kinematics when the path is complicated, because energy depends on states, not detailed motion along the path. That is one reason conservation of energy is so useful in mechanics.

Conservation of Energy and Power

Power measures how quickly energy is transferred or converted. The average power is

$$P = \frac{W}{t}$$

and instantaneous power can also be written as

$$P = \frac{dW}{dt}$$

Since work is a transfer of energy, power tells you the rate of energy transfer. A car engine, a human climbing stairs, or a motor lifting a box all involve power. Conservation of energy still holds, but power describes how fast energy changes happen.

For example, if a person gains gravitational potential energy by climbing stairs, the person’s body converts chemical energy into mechanical energy at a certain rate. A larger power output means the same energy change happens in less time. ⚡

Conclusion

Conservation of energy is one of the most important tools in AP Physics C: Mechanics. It explains how energy changes form while the total energy of an isolated system stays constant. In mechanics, you will often use the equation $K_i + U_i = K_f + U_f$ when only conservative forces act, or $K_i + U_i + W_{\text{nc}} = K_f + U_f$ when nonconservative forces are present.

students, if you remember one idea from this lesson, let it be this: energy methods let you compare the start and end of a physical process without needing every detail of the motion. That makes conservation of energy a powerful and efficient way to analyze motion, springs, ramps, pendulums, and many other systems.

Study Notes

Conservation of energy says the total energy of an isolated system stays constant.
Mechanical energy is usually $K + U$.
Kinetic energy is $K = \frac{1}{2}mv^2$.
Gravitational potential energy near Earth is $U_g = mgh$.
Spring potential energy is $U_s = \frac{1}{2}kx^2$.
If only conservative forces act, use $K_i + U_i = K_f + U_f$.
If nonconservative forces act, use $K_i + U_i + W_{\text{nc}} = K_f + U_f$.
Friction is nonconservative and turns mechanical energy into thermal energy.
Gravity and spring forces are conservative.
Choose a convenient zero level for gravitational potential energy.
Conservation of energy often gives faster solutions than kinematics.
Power describes the rate of energy transfer: $P = \frac{W}{t}$.
Energy problems are easiest when you clearly identify the system and the energies involved.