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Power is rate

Power is the rate of work or energy transfer:

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

Mechanical shortcut

For a constant force parallel to velocity,

$P=Fv$

Keep energy and power distinct

Energy is measured in joules; power is measured in watts, or joules per second. Multiply power by time to recover transferred energy.

Worked example from local Question Bank row 29322

A student of weight $600\,\mathrm{N}$ climbs $6.0\,\mathrm{m}$ vertically in $8.0\,\mathrm{s}$ .

$\Delta W=(600)(6.0)=3.6\times10^3\,\mathrm{J}$
$P=\frac{3.6\times10^3}{8.0}=4.5\times10^2\,\mathrm{W}=450\,\mathrm{W}$

The result is the average rate of energy transfer against gravity.

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A student of mass m initially at rest takes t seconds to run up stairs of height h. At the top of the stairs the student has a velocity v.

What is the average power supplied by the student during the climb?

Account for energy

Define the system, identify energy stores and describe transfers. Work done by a force transfers energy; total energy is conserved even when mechanical energy is not.

Use the mechanical model

$E_k= rac12mv^2,\quad \Delta E_{p,g}=mg\Delta h,\quad E_{p,elastic}= rac12k(\Delta x)^2$

Conserve their sum only when resistive transfers are absent or included explicitly.

Use rates and ratios

$P= rac{\Delta E}{\Delta t}=Fv,\qquad \eta= rac{E_{useful}}{E_{input}}= rac{P_{useful}}{P_{input}}$

Fuel energy density connects available input energy to a chosen volume.

Final checks

Check the system boundary, signs of work and potential-energy changes, the reference height, extension from natural length, and whether the quantity is energy, power, efficiency or energy density.

A.3.12—Power

Calculate Power as a Transfer Rate

Retrieve the A.3 Work, Energy and Power Model

A.3.12 Exam Analysis