Work, Energy, and Power: Fundamental Principles in A-Level Science

Defining Mechanical Work

Work is energy transfer via force application:
\[
W = \vec{F} \cdot \vec{d} = Fd\cos\theta
\]
Special Cases:

• \(\theta = 0^\circ\): \(W = Fd\) (maximum work)
• \(\theta = 90^\circ\): \(W = 0\) (no work done)

Energy Fundamentals

Kinetic Energy (K)

\[
K = \frac{1}{2}mv^2
\]
Relativistic Correction:
For \(v > 0.1c\):
\[
K = (\gamma – 1)mc^2 \quad \text{where} \quad \gamma = \frac{1}{\sqrt{1-(v/c)^2}}
\]

Potential Energy (U)

Gravitational:
\[
U_g = mgh
\]
Elastic (Spring):
\[
U_e = \frac{1}{2}kx^2
\]

Power and Efficiency

Instantaneous Power

\[
P = \frac{dW}{dt} = Fv
\]
Typical Values:

• Human climbing stairs: ~200W
• Car engine: 50-300 kW

System Efficiency

\[
\eta = \frac{\text{Useful Output}}{\text{Total Input}} \times 100\%
\]

Conservation Principles

Mechanical Energy:
\[
K_i + U_i = K_f + U_f \quad \text{(Closed systems)}
\]
Practical Limitations:

• 10-15% energy loss in mechanical systems
• 5-8% transmission loss in power grids

Practical Applications

Transportation Engineering

• Electric vehicles: 80-90% motor efficiency
• Regenerative braking recovers 15-25% energy

Energy Systems

• Wind turbines: 30-50% theoretical max (Betz limit)
• Solar panels: 15-22% typical efficiency

Worked Example

Lifting a 50kg mass 10m in 5s:

1. Work Done:
   \[
   W = mgh = 50 \times 9.81 \times 10 = 4,905 \, \text{J}
   \]
2. Power Required:
   \[
   P = \frac{W}{t} = \frac{4905}{5} = 981 \, \text{W}
   \]

Common Errors

1. Using average velocity in \(P = Fv\) calculations
2. Neglecting energy dissipation in conservation problems
3. Confusing spring potential (\( \frac{1}{2}kx^2 \)) with gravitational potential

Practice Problems

1. A 1,200kg electric car accelerates from 0 to 27m/s (60mph) in 6s. Calculate:
   • Final kinetic energy
   • Average power output
2. Derive the work-energy theorem (\(W_{net} = \Delta K\)) from Newton’s laws
3. Compare energy conversion efficiency in fossil fuel vs. electric vehicles.