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

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Leander Oates

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Physics

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:

$P_{i n s t} = F_{∥} v = F v \cos θ$

Where:
- $P_{i n s t}$ = instantaneous power, measured in $W$
- $F_{∥}$ = parallel component of applied force, measured in $N$
- $v$ = speed, measured in $m / s$
- $θ$ = 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:

$P_{i n s t} = F_{∥} v = F v \cos θ$

Derivation

Step 1: Identify the fundamental principle

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

$W = F_{∥} d = F d \cos θ$

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:

$P = \frac{F_{∥} d}{∆ t} = \frac{F d \cos θ}{∆ t}$

Recall that distance with respect to time is average speed

$v_{a v e} = \frac{d}{∆ t}$

Isolate average speed in the equation:

$P = F_{∥} \frac{d}{∆ t} = F \cos θ \frac{d}{∆ t}$

Substitute average speed into the equation:

$P = F_{∥} v = F v \cos θ$

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

$P_{i n s t} = F_{∥} v = F v \cos θ$

Worked Example

A student pushes a box of books along a wooden floor at a speed of $0.23 m / s$ from the hallway into the bedroom. The student provided a net force of $158 N$ at an angle of $45 °$ .

Which of the following is the instantaneous power delivered to the box?

A: $26 W$

B: $36 W$

C: $42 W$

D: $63 W$

The correct answer is A

Answer:

Step 1: List the known quantities

Speed, $v = 0.23 m / s$
Net force, $F = 158 N$
Angle of applied force at point of contact, $θ = 45 °$

Step 2: State the equation for instantaneous power

$P_{i n s t} = F v \cos θ$

Step 3: Substitute in the known values to calculate

$P_{i n s t} = (158) (0.23) (\cos (45))$

$P_{i n s t} = 26 W (2 s . f .)$

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

Study Guide

Instantaneous power

Derived equation

Derivation

Worked Example

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Author: Leander Oates