1.5: Power, Energy and Efficiency
- Page ID
- 98380
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Energy
Power (P) and energy (w) are interconnected concepts. Power is the rate at which energy is transferred or converted, and the relationship between power and energy involves calculus. The power (P) can be expressed as the derivative of energy (w) with respect to time (t), denoted as
\[ P(t) = \frac{dw}{dt} \]
Additionally, the total energy (wtotal) within a given time interval [t1 , t2] can be determined by integrating the power function over that interval.
\[ w_{\text{total}} = \int_{t_1}^{t_2} P(t) \, dt \]
Example: Finding Power with Given Energy
Assume the energy function w(t) is given as w(t)=3t2+2t+7 joules. The corresponding power function is obtained by taking the derivative:
\[P(t) = \frac{dw}{dt} = \frac{d}{dt}(3t^2 + 2t + 7)\ = 6t + 2\]
Evaluate the derivative and substitute a specific time, say t=2 seconds, to find the power at that moment.
Example: Determining Total Energy within t1 and t2:
Given a power function P(t)=6t+2, the total energy transferred between t1=1 second and t2=4 seconds is calculated by integrating the power function over the specified interval:
\[ w_{\text{total}} = \int_{t_1}^{t_2} (6t + 2) \, dt = 3t^2 + 2t + c|^{4}_{1} = 3(4)^2 + 2(4) + c - 3(1)^2 - 2(1) - c = 51 joules \]
Evaluate the definite integral to find the total energy transferred during this period.