Instantaneous Power (College Board AP® Physics 1: Algebra-Based): Revision Note

Instantaneous power

• Instantaneous power is the power delivered to an object in an exact instant in time

• The instantaneous power delivered to an object by the component of a constant force parallel to the object’s velocity can be described by the following equation:

$P_{inst}=F_{\parallel }v=Fv\cos \theta$

• Where:

  • $P_{inst}$ = instantaneous power, measured in $\mathrm{W}$

  • $F_{\parallel }$ = parallel component of applied force, measured in $\mathrm{N}$

  • $v$ = speed, measured in $\mathrm{m}/\mathrm{s}$

  • $\theta$ = angle of applied force at point of contact, measured in $°$

Derived equation

• For an object being displaced by an applied force, the instantaneous power is given by the equation:

$P_{inst}=F_{\parallel }v=Fv\cos \theta$

Derivation

Step 1: Identify the fundamental principle

• The work done to displace an object by an applied force is given by:

$W=F_{\parallel }d=Fd\cos \theta$

Step 2: Apply the specific conditions

• Power is the work done (or energy transferred) with respect to time

$P=\frac{W}{∆t}$

• Substitute the work equation into the power equation:

$P=\frac{F_{\parallel }d}{∆t}=\frac{Fd\cos \theta }{∆t}$

• Recall that distance with respect to time is average speed

$v_{ave}=\frac{d}{∆t}$

• Isolate average speed in the equation:

$P=F_{\parallel }\frac{d}{∆t}=F\cos \theta \frac{d}{∆t}$

• Substitute average speed into the equation:

$P=F_{\parallel }v=Fv\cos \theta$

• This equation gives the power delivered by a specific force displacing an object at a specific speed; therefore, it is instantaneous power

$P_{inst}=F_{\parallel }v=Fv\cos \theta$