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Analyze Energy Using
Conservation Laws

Master the fundamental principle that energy cannot be created or destroyed: it can only transform from one form to another. Apply \( E = KE + PE = \text{constant} \) to solve real-world physics problems.

High School Physics

What is Energy?

Energy is the capacity to do work on an object. In physics, we measure energy in joules (J). Every physical process involves energy changing form: a ball rolling downhill, a spring compressing, or a pendulum swinging.

Kinetic Energy
Energy of motion: \( KE = \frac{1}{2}mv^2 \)
Potential Energy
Stored energy due to position: \( PE = mgh \)
Mechanical Energy
Total: \( E = KE + PE \)

Forms of Energy in Physics

Kinetic Moving objects Gravitational PE Height above ground Elastic PE Springs, rubber bands Mechanical Energy
\( E = KE + PE \)
 Energy transforms between these forms but the total stays constant

Why Conservation Laws Matter

Conservation of energy is one of the most powerful tools in all of physics. Instead of tracking forces and accelerations at every instant, you compare energy at two points and solve directly. It works for roller coasters, pendulums, projectiles, and springs.

1
Simplifies complex problems: No need for force diagrams along curved paths.
2
Works in any direction: Energy is a scalar: no vector components needed.
3
Universal principle: Applies to everything from atoms to galaxies.

The Conservation of Mechanical Energy

h=0 A PE: high KE: low B PE: low KE: high C PE: highest KE: lowest
\( h_1 \)
\( h_2 \)
 A: KE+PE B: KE+PE C: KE+PE

When Non-Conservative Forces Act

Friction, air resistance, and applied pushes are non-conservative forces. They add or remove energy from the system. The generalized conservation equation becomes:

Kinetic vs Potential Energy

Kinetic Energy Potential Energy
Formula \( \frac{1}{2}mv^2 \) \( mgh \) (gravitational)
Depends on Mass and speed Mass and height
When maximum At lowest point At highest point
When zero Object at rest At reference level \( h = 0 \)
Scalar / Vector Scalar (always \( \ge 0 \)) Scalar (can be \( < 0 \) if below ref.)

Elastic Potential Energy

Springs store energy when compressed or stretched. The elastic potential energy depends on how far the spring is displaced from equilibrium:

\(m\)
\( x = 0 \)
\( x \)

How to Apply Conservation of Energy

1
Identify the system: What object(s) are you tracking? Is the system isolated?
2
Choose two states: Pick an initial and final position (e.g., top and bottom of a hill).
3
Set a reference level: Define where \( h = 0 \) for gravitational PE.
4
Write the energy equation: \( KE_i + PE_i + W_{\text{nc}} = KE_f + PE_f \)
5
Substitute and solve: Plug in known values, cancel zeros, solve for the unknown.

Example 1: Falling Ball

A 2 kg ball is dropped from rest at height \( h = 10 \) m. Find its speed just before hitting the ground. (Ignore air resistance.)

h=0 10 m m v = 0
\( PE = mgh \)
 m v = ?
\( KE = \frac{1}{2}mv^2 \)

Example 2: Roller Coaster

A 500 kg roller coaster car starts from rest at height \( h_1 = 40 \) m. What is its speed at \( h_2 = 15 \) m? (No friction.)

h=0 A 40 m B 15 m A: all PE B: KE+PE

Example 3: Spring Launch

A 0.5 kg ball rests on a vertical spring (\( k = 800 \text{ N/m} \)) compressed by \( x = 0.1 \) m. How high does the ball rise after release?

Example 4: Friction on a Slide

A 10 kg child slides down a 3 m high slide. Friction does \( W_f = -100 \text{ J} \) of work. Find the speed at the bottom.

3 m Start
\( v=? \)
 friction h=0

Reading Energy Bar Charts

Energy bar charts visually show how energy transforms between states. Each bar represents one form of energy. The total height of all bars stays the same (conservation).

Initial State KE PE Total E → Final State KE PE Total E

Common Mistakes

✕ Forgetting Reference Level
You must define \( h = 0 \) before calculating PE. Different reference levels give different PE values but the same answer for \( \Delta PE \).
✕ Confusing Energy and Forces
Energy methods are great for finding speed, but not for finding the time taken or the exact force at a point on a curve. Use kinematics or Newton's laws for those.
✕ Ignoring Non-Conservative Work
If friction or an applied force acts, you must include \( W_{\text{nc}} \) in the equation. Leaving it out gives the wrong answer.

Summary of Energy Equations

Scenario Equation When to Use
No friction \( KE_i + PE_i = KE_f + PE_f \) Gravity only, no air resistance
With friction \( KE_i + PE_i + W_{\text{nc}} = KE_f + PE_f \) Friction, applied forces present
Spring problems \( \frac{1}{2}kx^2 + mgh = \frac{1}{2}mv^2 + mgh' \) Elastic PE involved
Work-energy theorem \( W_{\text{net}} = \Delta KE \) Finding net work or speed change

Question 1 of 5

A 4 kg ball is dropped from a height of 20 m. What is its speed just before hitting the ground? (use \( g = 9.8 \text{ m/s}^2 \))

Question 2 of 5

A roller coaster starts from rest at 50 m and reaches a second hill at 30 m. What is its speed at the top of the second hill? (No friction, use \( g = 9.8 \text{ m/s}^2 \))

Question 3 of 5

A spring with \( k = 200 \text{ N/m} \) is compressed by 0.3 m. How much elastic potential energy is stored?

Question 4 of 5

A 5 kg box slides down a 4 m high ramp. Friction does \( -60 \text{ J} \) of work. What is the box's speed at the bottom? (use \( g = 9.8 \text{ m/s}^2 \))

Question 5 of 5

A pendulum is released from a height of 0.8 m above its lowest point. What is its speed at the lowest point? (use \( g = 9.8 \text{ m/s}^2 \))