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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 3: Work, Energy & PowerTranslational Kinetic Energy & Potential EnergyGravitational Potential Energy Near a Surface

Gravitational Potential Energy Near a Surface (College Board AP® Physics 1: Algebra-Based)

Study Guide

Leander Oates

Written by: Leander Oates

Reviewed by: Caroline Carroll

Gravitational potential energy near a surface

  • The gravitational field near the surface of a planet is nearly constant

  • Therefore, it can be approximated to be constant

Diagram depicting gravitational field lines around a planet. Radial field lines point towards the center, while lines near the surface are parallel and uniform.
The gravitational field at the surface of a planet is almost uniform, therefore it can be approximated as being uniform

  • The change in gravitational potential energy in a system consisting of an object near the surface of a planet and a planet, can be approximated by the equation:

increment U subscript g space equals space m g increment y

  • Where:

    • increment U subscript g = change in gravitational potential energy, measured in straight J

    • m = mass of object moving through a gravitational field, measured in kg

    • g = gravitational field strength of planet, measured in straight N divided by kg

    • increment y = vertical distance moved by object through a gravitational field, measured in straight m

  • This equation assumes that the gravitational field through which the object moves is constant

  • The gravitational potential energy and the height through which the object is moved are relative

    • The zero points of potential energy and displacement are adjusted based on the situation (this is covered in greater detail in the study guide on potential energy)

    • This is shown by the increment (delta) symbol, meaning change in

  • When an object is lifted through a gravitational field, the system gains gravitational potential energy

  • When an object is lowered through a gravitational field, the system loses gravitational potential energy

Diagram showing an object thrown in the air and an object falling, with corresponding graphs of gravitational potential energy (GPE) against height for each scenario.
Graphs showing the linear relationship between gravitational potential energy and height

  • The gravitational force is a conservative force; therefore, gravitational potential energy is path-independent

  • Gravitational potential energy is a scalar quantity with magnitude only

Worked Example

A rock of mass 2.1 space kg is dropped from a cliff into the water 12 space straight m below. The person that drops the rock releases it from a height of 1.4 space straight m above the ground. Which of the following values is correct for the speed of the rock just before it hits the water?

A: 6.5 space straight m divided by straight s

B: 14.5 space straight m divided by straight s

C: 15.5 space straight m divided by straight s

D: 16.4 space straight m divided by straight s

The correct answer is D

Step 1: Analyze the scenario

  • The cliff is 12 space straight m above the water, but the rock is dropped from a height of 1.4 space straight m above the ground; therefore, the total vertical distance the rock travels is 12 space plus space 1.4 space equals space 13.4 space straight m

  • Air resistance is always assumed to be negligible unless otherwise stated

  • Therefore, it can be assumed that the gravitational potential energy is transformed into kinetic energy with 100% efficiency

Step 2: List the known quantities

  • Mass of rock, m space equals space 2.1 space kg

  • Vertical distance traveled by rock, increment y space equals space 13.4 space straight m

  • Magnitude of the gravitational field strength at the Earth's surface, g space equals space 10 space straight N divided by kg

Step 3: Calculate the change in gravitational potential energy as the rock falls to the surface of the water

increment U subscript g space equals space m g increment y

increment U subscript g space equals space 2.1 space times space 10 space times space 13.4 space equals space 281.4 space straight J

Step 4: Equate the change in gravitational potential energy to the gain in kinetic energy

increment U subscript g space equals space K space equals space 1 half m v squared

Step 5: Rearrange to make speed the subject and substitute in the known values to calculate

increment U subscript g space equals space 1 half m v squared space space rightwards double arrow space space v space equals space square root of fraction numerator 2 times space increment U subscript g over denominator m end fraction end root

v space equals space square root of fraction numerator 2 space times space 281.4 over denominator 2.1 end fraction end root space equals space 16.4 space straight m divided by straight s

Examiner Tips and Tricks

You could also solve this problem by equating the expressions for gravitational potential energy and translational kinetic energy:

m g increment y space equals space 1 half m v squared

g increment y space equals space 1 half v squared

v space equals space square root of 2 g increment y end root

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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.