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Conservation of
Energy

Master the fundamental principle that energy cannot be created or destroyed, only transformed from one form to another.

AP Physics 1

What Is Energy?

Energy is the capacity to do work or cause change. In physics, it is a scalar quantity measured in joules (J). Energy exists in many forms, and one of the most powerful ideas in all of physics is that the total energy of an isolated system remains constant.

Kinetic Energy
Energy of motion: \( KE = \tfrac{1}{2}mv^2 \)
Potential Energy
Stored energy due to position or configuration

Key Energy Types in Mechanics

Kinetic
\( KE = \tfrac{1}{2}mv^2 \)
Depends on mass and speed
Gravitational PE
\( U_g = mgh \)
Depends on height above reference
Spring PE
\( U_s = \tfrac{1}{2}kx^2 \)
Depends on compression or stretch

Work and Energy Transfer

Work is the mechanism by which energy is transferred into or out of a system. When a net force acts on an object through a displacement, it does work and changes the object's kinetic energy.

The Conservation of Energy

In an isolated system (no external work), the total mechanical energy is conserved. If only conservative forces do work:

Visualizing Energy with Bar Charts

Top of Hill KE PE Total E transforms Bottom of Hill KE PE Total E KE PE Total

Conservative vs Non-Conservative Forces

Conservative Non-Conservative
Definition Work is path-independent Work depends on path taken
Examples Gravity, springs Friction, air resistance
Has a PE function? Yes No
Effect on total \( E_{\text{mech}} \) Conserves it Decreases it (converts to thermal)
Round-trip work \( W = 0 \) \( W \neq 0 \)

How to Apply Conservation of Energy

1
Define the system. Identify what objects and forces are included. Choose a reference level for \( h = 0 \).
2
Identify initial and final states. What are the positions and speeds at each point?
3
Check for non-conservative forces. Is friction or air resistance present? If so, include \( W_{\text{nc}} \).
4
Write the energy equation: \( KE_i + U_i + W_{\text{nc}} = KE_f + U_f \)
5
Substitute and solve. Plug in known values and solve for the unknown.

Example 1: Falling Ball

A 2 kg ball is dropped from rest at a height of 10 m. What is its speed just before hitting the ground? (Ignore air resistance.)

ground (\(h = 0\))
\(h=10\text{ m}\)
2 kg
\(v=0\)
2 kg
\(v=?\)

Example 2: Roller Coaster Loop

A 500 kg cart starts from rest at the top of a 40 m hill and enters a circular loop of radius 10 m. What is its speed at the top of the loop? (No friction.)

\(v=0\)
\(40\text{ m}\)
\(v=?\)
\(20\text{ m}\)

Example 3: Sliding with Friction

A 4 kg block slides down a 5 m ramp inclined at \(30^\circ\). Friction does \(-40 \text{ J}\) of work on the block. If it starts from rest, what is its speed at the bottom?

\(h\)
\(5\text{ m}\)
\(30^\circ\)
friction
\(v=?\)

Example 4: Spring Launcher

A spring with \( k = 800 \) N/m is compressed 0.15 m and launches a 0.5 kg ball vertically upward. How high does the ball rise?

0.5 kg
\(x=0.15\text{ m}\)
0.5 kg
\(h=?\)

Energy and Power

Power is the rate at which energy is transferred or transformed. It tells you how quickly work is done, not how much.

Average Power
\( P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t} \)
Instantaneous Power
\( P = Fv\cos\theta \)

Common Mistakes to Avoid

✕ Forgetting the Reference Level
Always define where \( h = 0 \). Height is relative: changing your reference changes individual PE values, but energy differences stay the same.
✕ Ignoring Non-Conservative Work
If friction is present, you cannot write \( KE_i + U_i = KE_f + U_f \). You must include \( W_{\text{nc}} \), which is always negative for friction.
✕ Confusing Force with Energy
Energy is scalar (no direction). Students sometimes try to break energy into components like forces. Don't: just add up the energy terms.

Summary of Energy Equations

Quantity Formula Units
Kinetic Energy \( \tfrac{1}{2}mv^2 \) J
Gravitational PE \( mgh \) J
Spring PE \( \tfrac{1}{2}kx^2 \) J
Conservation (no friction) \( KE_i + U_i = KE_f + U_f \) J
Conservation (with friction) \( KE_i + U_i + W_{\text{nc}} = KE_f + U_f \) J
Power \( P = \frac{\Delta E}{\Delta t} \) W

Question 1 of 5

A ball is thrown straight up with an initial speed of 20 m/s. Ignoring air resistance, what is its speed when it returns to the same height?

Question 2 of 5

A 3 kg object slides down a frictionless ramp from a height of 5 m. What is its kinetic energy at the bottom?

Question 3 of 5

A spring with \( k = 200 \) N/m is compressed 0.1 m. How much elastic potential energy is stored in the spring?

Question 4 of 5

A block slides across a rough surface and slows down. Which statement is true about the system's energy?

Question 5 of 5

Two identical balls are released from the same height: one rolls down a straight ramp, the other along a curved valley. Ignoring friction, which reaches the bottom with greater speed?