Topic 3.5: Power

Name: _______________________________ Date: _______________ Period: _______

Before We Start: Yesterday's Recap

Two students each carry a $10\text{ kg}$ box up the same flight of stairs — Student A in 10 s, Student B in 20 s. Who does more work against gravity? Who was more powerful? What's the difference?

Today's New Concepts

AP CED Alignment: Unit 3, Topic 3.5

Average Power: $P=W/t$

Power is the rate of energy transfer: $P_{avg}=\dfrac{\Delta E}{\Delta t}=\dfrac{W}{\Delta t}=\dfrac{Fd}{\Delta t}$ (force parallel to displacement)
Unit: watt, $1\text{ W}=1\text{ J/s}$. (Horsepower: $1\text{ hp}\approx746\text{ W}$.)
Same work in less time $\Rightarrow$ more power

Worked Example: A $60\text{ kg}$ student climbs $4.0\text{ m}$ of stairs in $8.0\text{ s}$: $W=mgh=2400\text{ J}$, $P_{avg}=2400/8.0=300\text{ W}$.

Instantaneous Power: $P=Fv$

For a force parallel to velocity: $P=Fv$ (from $\dfrac{Fd}{t}=F\dfrac{d}{t}$)
Constant speed up a frictionless ramp: $F=mg\sin\theta$, so $P=mg\sin\theta;v$ — steeper ramp, same speed $\Rightarrow$ more power (more force needed)

Worked Example: A $2.0\text{ kg}$ block is pulled at $1.5\text{ m/s}$ up a frictionless $30°$ ramp: $P=(2.0)(10)(0.50)(1.5)=15\text{ W}$; at $45°$: $\approx21\text{ W}$.

Power From Kinematics, and Power-vs-Time Graphs

Constant acceleration from rest over distance $d$ in time $t$: $a=\dfrac{2d}{t^2}$, $W=mad$, $P_{avg}=\dfrac{2md^2}{t^3}$. From $\Delta K/\Delta t$: $P_{avg}=\dfrac{mv^2}{2t}$
Constant force from rest: the final instantaneous power is twice the average power
Area under a power-vs-time graph = energy transferred. Different graph shapes can have equal area (equal energy)

Worked Example: Graph 1: $P$ falls from $6\text{ W}$ to $2\text{ W}$ over $4\text{ s}$. Graph 2: $P$ rises from $2\text{ W}$ to $6\text{ W}$ over $4\text{ s}$. Both areas $=\tfrac12(6+2)(4)=16\text{ J}$, so $\Delta E_1=\Delta E_2$.