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Work, Energy, and Power: Fundamental Principles in A-Level Science
Defining Mechanical Work
Work is energy transfer via force application:
![\[ W = \vec{F} \cdot \vec{d} = Fd\cos\theta \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-d24a54a25aa49ff3d7e0e3c23ba1136c_l3.png "Rendered by QuickLaTeX.com")
Special Cases:
: 
(maximum work)
: 
(no work done)
: 
(maximum work)
: 
(no work done)Energy Fundamentals
Kinetic Energy (K)
![\[ K = \frac{1}{2}mv^2 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-14dee2ac6a497cbc88ab012f1105b999_l3.png "Rendered by QuickLaTeX.com")
Relativistic Correction:
For 
:
![\[ K = (\gamma - 1)mc^2 \quad \text{where} \quad \gamma = \frac{1}{\sqrt{1-(v/c)^2}} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-573e8d37b816456105a1c23aedeaac67_l3.png "Rendered by QuickLaTeX.com")
Potential Energy (U)
Gravitational:
![\[ U_g = mgh \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-ccbbf0f00c4be3d618caab94c13e52bd_l3.png "Rendered by QuickLaTeX.com")
Elastic (Spring):
![\[ U_e = \frac{1}{2}kx^2 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-141420d416582738ce5a2dc34123e3cb_l3.png "Rendered by QuickLaTeX.com")
Power and Efficiency
Instantaneous Power
![\[ P = \frac{dW}{dt} = Fv \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-e6cb52aeda51f160967e3bc73eb5d73f_l3.png "Rendered by QuickLaTeX.com")
Typical Values:
- Human climbing stairs: ~200W
- Car engine: 50-300 kW
System Efficiency
![\[ \eta = \frac{\text{Useful Output}}{\text{Total Input}} \times 100\% \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-8073a3b5a1cc4c988d512ca9db75a4fc_l3.png "Rendered by QuickLaTeX.com")
Conservation Principles
Mechanical Energy:
![\[ K_i + U_i = K_f + U_f \quad \text{(Closed systems)} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-ec8aee84f0e098d7825a5bc51cdadb7a_l3.png "Rendered by QuickLaTeX.com")
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:
- Work Done:
![\[ W = mgh = 50 \times 9.81 \times 10 = 4,905 \, \text{J} \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOTEiIGhlaWdodD0iMTciIHZpZXdCb3g9IjAgMCAyOTEgMTciPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Power Required:
![\[ P = \frac{W}{t} = \frac{4905}{5} = 981 \, \text{W} \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTUiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxOTUgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
![\[ W = mgh = 50 \times 9.81 \times 10 = 4,905 \, \text{J} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-49b13519e757aa8e911cba8885f2e020_l3.png "Rendered by QuickLaTeX.com")
![\[ P = \frac{W}{t} = \frac{4905}{5} = 981 \, \text{W} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-7c8e2ccc2f2523a0f998dad698799ede_l3.png "Rendered by QuickLaTeX.com")
Common Errors
- Using average velocity in
calculations - Neglecting energy dissipation in conservation problems
- Confusing spring potential (
) with gravitational potential
calculations
) with gravitational potentialPractice Problems
- A 1,200kg electric car accelerates from 0 to 27m/s (60mph) in 6s. Calculate:
- Final kinetic energy
- Average power output
- Derive the work-energy theorem (
) from Newton’s laws - Compare energy conversion efficiency in fossil fuel vs. electric vehicles
) from Newton’s lawsSkinat Tuition | Empowering Global Academic Success, One Student at a Time.