AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 3: Work, Energy & PowerPowerInstantaneous Power

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

Study Guide

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Leander Oates

Written by: Leander Oates

Reviewed by: Caroline Carroll

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 subscript i n s t end subscript space equals space F subscript parallel to v space equals space F v space cos space theta

  • Where:

    • P subscript i n s t end subscript = instantaneous power, measured in straight W

    • F subscript parallel to = parallel component of applied force, measured in straight N

    • v = speed, measured in straight m divided by straight s

    • theta = angle of applied force at point of contact, measured in degree

Derived equation

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

P subscript i n s t end subscript space equals space F subscript parallel to v space equals space F v space cos space theta

Derivation

Step 1: Identify the fundamental principle

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

W space equals space F subscript parallel to d space equals space F d space cos theta

Step 2: Apply the specific conditions

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

P space equals space fraction numerator W over denominator increment t end fraction

  • Substitute the work equation into the power equation:

P space equals space fraction numerator F subscript parallel to d over denominator increment t end fraction space equals space fraction numerator F d space cos space theta over denominator increment t end fraction

  • Recall that distance with respect to time is average speed

v subscript a v e end subscript space equals space fraction numerator d over denominator increment t end fraction

  • Isolate average speed in the equation:

P space equals space F subscript parallel to fraction numerator d over denominator increment t end fraction space equals space F space cos space theta fraction numerator d over denominator increment t end fraction

  • Substitute average speed into the equation:

P space equals space F subscript parallel to v space equals space F v space cos space theta

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

P subscript i n s t end subscript space equals space F subscript parallel to v space equals space F v space cos space theta

Worked Example

A student pushes a box of books along a wooden floor at a speed of 0.23 space straight m divided by straight s from the hallway into the bedroom. The student provided a net force of 158 space straight N at an angle of 45 degree.

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

A: 26 space straight W

B: 36 space straight W

C: 42 space straight W

D: 63 space straight W

The correct answer is A

Answer:

Step 1: List the known quantities

  • Speed, v space equals space 0.23 space straight m divided by straight s

  • Net force, F space equals space 158 space straight N

  • Angle of applied force at point of contact, theta space equals space 45 degree

Step 2: State the equation for instantaneous power

P subscript i n s t end subscript space equals space F v space cos space theta

Step 3: Substitute in the known values to calculate

P subscript i n s t end subscript space equals space open parentheses 158 close parentheses open parentheses 0.23 close parentheses open parentheses cos open parentheses 45 close parentheses close parentheses

P subscript i n s t end subscript space equals space 26 space straight W space open parentheses 2 space straight s. straight f. close parentheses

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    Author
    Reviewer
    Leander Oates

    Author: Leander Oates

    Expertise: Physics

    Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.

    Caroline Carroll

    Author: Caroline Carroll

    Expertise: Physics Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.