Conservation of Energy

Mechanical Energy: This is the sum of kinetic and potential energies ($E_{mech} = KE + PE$). In the absence of non-conservative forces like friction, mechanical energy is conserved: $KE_i + PE_i = KE_f + PE_f$.

Work-Energy Theorem: Work is the mechanism of energy transfer. When an external force $F$ acts over a displacement $x$, the work done ($W = Fx$) is equal to the change in the system's energy.

Mathematical Models: Kinetic energy is defined as $KE = \frac{1}{2}mv^2$, while Gravitational Potential Energy near Earth's surface is $GPE = mgh$. Elastic Potential Energy in a spring is $EPE = \frac{1}{2}k(\Delta L)^2$.

Step 1: Define the System and States: Clearly identify the 'Initial' and 'Final' moments of interest. Ensure the system boundaries are set so that all relevant energy forms are accounted for.

Step 2: Establish a Reference Level: For potential energy, choose a height where $h = 0$ (usually the lowest point in the problem). This simplifies calculations by making one term zero.

Step 3: Account for Resistive Forces: If friction or air resistance is present, use the expanded energy balance: $E_{initial} = E_{final} + W_{friction}$. The work done against friction is usually calculated as $W = f \cdot d$, where $d$ is the path length.

Step 4: Solve for the Unknown: Substitute the known formulas into the balance equation and solve algebraically for the target variable, such as velocity or height.

The 'Height' Trap: In GPE calculations ($mgh$), $h$ is always the vertical displacement, not the distance traveled along a slope. Use trigonometry ($L \sin heta$) to find the vertical component if only the slope length is given.

Velocity Squared: Remember that kinetic energy depends on $v^2$. If the speed of an object doubles, its kinetic energy quadruples ($2^2 = 4$). This is a frequent source of calculation errors in multi-step problems.

Sanity Checks: Always check if the final energy is less than or equal to the initial energy. If your final velocity is higher than what conservation allows, you likely missed a resistive force or made an algebraic error.

Unit Consistency: Ensure all masses are in kilograms (kg), heights in meters (m), and speeds in meters per second (m/s) to ensure the resulting energy is in Joules (J).

Energy is Not a Vector: Unlike force or momentum, energy is a scalar quantity. It does not have a direction. When summing energies, you do not need to resolve them into x and y components; you simply add the magnitudes.

'Lost' Energy: Students often say energy is 'lost' to friction. In physics, energy is never lost; it is simply converted into Internal (Thermal) Energy, which is harder to utilize for mechanical work.

Spring Compression: For Elastic Potential Energy, the $\Delta L$ in $\frac{1}{2}k(\Delta L)^2$ is the change in length from the equilibrium position, not the total length of the spring.