For many problems:

$E_{mech} = \frac{1}{2}mv^2 + mgh + \frac{1}{2}kx^2$

(kinetic + gravitational PE + elastic PE)

Conservation of Mechanical Energy

When only conservative forces do work:

$E_i = E_f$

$KE_i + PE_i = KE_f + PE_f$

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

💡 Key: Total mechanical energy stays constant - energy just transforms between KE and PE!

Conservative vs. Non-Conservative Forces

Conservative Forces

Forces for which mechanical energy is conserved:

• Gravity - can define gravitational PE
• Spring force - can define elastic PE
• Electrostatic force - can define electric PE

Properties:

• Work is path-independent (depends only on start/end points)
• Potential energy can be defined
• Work around closed path = 0

Non-Conservative Forces

Forces that dissipate mechanical energy:

• Friction - converts mechanical energy to thermal energy
• Air resistance - converts KE to thermal energy
• Tension (usually) - can do positive or negative work

Important: When non-conservative forces do work, mechanical energy is NOT conserved!

Energy with Non-Conservative Forces

When friction or other non-conservative forces are present:

$E_i + W_{nc} = E_f$

$KE_i + PE_i + W_{nc} = KE_f + PE_f$

where $W_{nc}$ = work done by non-conservative forces

Special case: Friction

$W_{friction} = -f_k \cdot d = -\mu_k N \cdot d$

(Negative because friction opposes motion)

This energy is converted to thermal energy (heat) and "lost" from mechanical energy.

Problem-Solving with Energy Conservation

Step-by-Step Strategy

1. Identify the system and choose reference points
2. Choose two points in the motion (initial and final)
3. List energies at each point:
   • Kinetic: $\frac{1}{2}mv^2$
   • Gravitational PE: $mgh$
   • Elastic PE: $\frac{1}{2}kx^2$
4. Check for non-conservative forces
5. Write conservation equation:
   • If only conservative forces: $E_i = E_f$
   • If non-conservative forces present: $E_i + W_{nc} = E_f$
6. Solve for unknown

Common Energy Transformations

Pendulum

• At highest points: Maximum PE, minimum KE (zero if released from rest)
• At lowest point: Minimum PE, maximum KE
• Continuous conversion: $PE \leftrightarrow KE$

Roller Coaster

• Top of hill: High PE, low KE
• Bottom of dip: Low PE, high KE
• First hill must be highest (no external energy added)

Vertical Spring

• Maximum compression: Maximum elastic PE
• Equilibrium: Maximum KE
• Includes both elastic and gravitational PE!

Projectile Motion

• At launch: Some KE, some PE (if not ground level)
• At peak: Horizontal KE only, maximum PE
• At landing: Maximum KE (if returns to launch height)

Energy Bar Diagrams

A useful visualization tool showing energy distribution. Energy bar diagrams show the distribution of kinetic and potential energy:

• Initial state: Mostly potential energy (at height)
• Final state: Mostly kinetic energy (at bottom)
• Total bar height stays same if energy conserved

⚠️ Common Mistakes

Mistake 1: Forgetting Energy Types

Don't forget to include ALL forms of energy! If there's a spring, include elastic PE. If there's height, include gravitational PE.

Mistake 2: Wrong Reference Point

Choose a consistent reference point for PE. Usually pick the lowest point as $h = 0$.

Mistake 3: Ignoring Non-Conservative Work

If friction is present, you CANNOT use $E_i = E_f$. Must account for work by friction!

Mistake 4: Sign Errors

• PE increases as height increases (positive work against gravity)
• Friction work is negative (opposes motion)
• Spring PE is always positive

When to Use Energy vs. Force Methods

Use Energy Methods When:

• Asked for speed/velocity (not acceleration)
• Motion involves height changes
• Springs are involved
• Path is complicated but you only care about start/end

Use Force Methods (Newton's Laws) When:

• Asked for acceleration or force
• Need to find motion at specific point (not just start/end)
• Need to analyze contact forces

Many problems can be solved either way!

Power: Rate of Energy Transfer

Power is how fast energy is transferred:

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

Also: $P = Fv$ (for constant force)

Units: Watt (W) = J/s

1 horsepower (hp) = 746 W

Applications

Hydroelectric Power

Gravitational PE of water → KE → Electrical energy

Regenerative Braking

KE of car → Electrical energy (charges battery)

Bungee Jumping

Gravitational PE → KE → Elastic PE (of cord) → KE → Gravitational PE

Earthquakes

Elastic PE stored in rock → KE of seismic waves → Damage

Key Formulas Summary

Find: Final speed $v_f$

Step 1: Choose reference point

Let ground be $h = 0$.

Step 2: Calculate initial energy

At height 5 m:

• $KE_i = \frac{1}{2}mv_i^2 = 0$ (at rest)
• $PE_i = mgh_i = (2)(9.8)(5) = 98$ J

Total: $E_i = 98$ J

Step 3: Calculate final energy

At ground:

• $KE_f = \frac{1}{2}mv_f^2$ (unknown)
• $PE_f = mgh_f = 0$ J

Total: $E_f = \frac{1}{2}mv_f^2$

Step 4: Apply conservation of energy

$E_i = E_f$

$98 = \frac{1}{2}(2)v_f^2$

$98 = v_f^2$

$v_f = \sqrt{98} = 7\sqrt{2} \approx 9.90 \text{ m/s}$

Check with kinematics:

$v_f^2 = v_i^2 + 2g\Delta h = 0 + 2(9.8)(5) = 98$

$v_f = \sqrt{98}$ ✓

Answer: The speed just before hitting the ground is approximately 9.90 m/s or $7\sqrt{2}$ m/s.

Note: The mass cancels out - all objects fall at the same rate (ignoring air resistance)!