1. The Energy Audit
The Master Equation allows you to track energy from the Start (Initial) to the End (Final). Account for outside interference using External Work.
KEi + GPEi + EPEi + Wext = KEf + GPEf + EPEf
➕ Aiding Motion (Positive Work)
Forces pointing the same way as movement add energy to the system.
➖ Opposing Motion (Negative Work)
Forces pointing opposite to movement (like Friction) remove energy.
2. The Players
Understanding the individual components of our system. Each type of energy has a specific formula and logical role in the conservation cycle.
Kinetic (KE)
½mv²
The energy of motion. If the object stops moving, this bank account drops to zero.
Gravitational (GPE)
mgh
Energy of height. The higher you go relative to your 'zero' point, the more energy you store.
Elastic (EPE)
½kx²
Energy stored in a spring. Because distance (x) is squared, triple the stretch means 9x the energy!
Work (W)
F · Δx_parallel
Energy transfer. Only the force parallel to the motion counts towards changing the system's energy.
3. The Friction Workflow
Friction is the most common mechanism for energy removal. Follow this organized sequence to calculate the exact energy lost to the environment.
Find Normal Force
F_N
On a flat surface: mg. On a ramp: mg × 0.8.
Find Kinetic Friction
f_k = μ_k · F_N
Multiply normal force by the coefficient of friction.
Calculate Work
W = -(f_k) · Δx
Always negative! It opposes the direction of travel.
4. Geometry Shortcut & Energy Audit
Let's put the math to the test. First, we establish our geometric shortcuts, and then we analyze the comprehensive "Auk-a-pult" problem.
The 3-4-5 Geometry Shortcut
When a ramp is angled at 36.87°, avoid calculator lag by using the 3-4-5 triangle ratios for force components.
- sin(36.87°) = 0.6 (The "3" side - vertical/gravity)
- cos(36.87°) = 0.8 (The "4" side - normal force)
Case Study: The "Auk-a-pult"
A 2kg block is launched by a spring (k=500), crosses a 2m rough patch, and goes up a smooth ramp. Where does the energy go?
Initial: 40J of Spring Energy.
Friction: Steals 10J during the rough patch.
Final: The remaining 30J converts to 1.5m of height.