# Potential Energy of Gravity

by Ron Kurtus (revised 21 January 2011)

An object held above the ground has the potential of accelerating downward, due to the pull of gravity. In other words, the object has potential energy (PE) that can be turned into the kinetic energy (KE) of motion. The sum of the potential energy and kinetic energy due to gravity for an object is constant unless outside forces come into play.

The potential energy of an object increases with height off the ground. This means that dropping an object from greater heights results in higher velocity or kinetic energy. However, since g decreases with height, there is a limit to the increase of PE.

You can calculate the PE, KE and total energy (TE) for an object that is dropped, thrown downward or projected upward with some simple equations. You can then verify that the final velocity is the same as obtained from the gravity derivations.

Questions you may have include:

• What is the potential energy equation?
• What is the derivation of the potential energy equation?
• What is the final velocity of an object dropped from some height?

This lesson will answer those questions. Useful tool: Units Conversion

## Potential energy equation

The equation for the potential energy of an object due to gravity is:

PE = mgh

where

• PE is the potential energy in joules (J) or foot-pounds (ft-lbs)
• m is the mass of the object in kg-mass or pound-mass
• g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
• h is the height above the ground in m or ft

Note: Potential energy is also sometimes abbreviated as U.

Note: Although g is often stated as 32 ft/s2, using the English system of measurement for energy is awkward, because PE is stated in ft-lb-weight and mass m is stated lb-mass. Mixing weight and mass values can result in confusing calculations.

## Derivation of potential energy equation

This equation can be derived from the fact that the force of gravity is defined as the change in potential energy with a change in distance:

F = dPE/dh

where

• F is the force of gravity
• dPE is the derivative of PE
• dPE/dh is the change of PE with respect to the change in h

The equation can be rearranged as:

dPE = Fdh

Since F = mg:

dPE = (mg)dh

Assuming the potential energy is zero when the object is on the ground, you can use calculus to integrate dPE between the initial position when PE = PE and the final position when PE = 0. You can also integrate dh from the initial height h = h to the final height on the ground h = 0. Thus:

∫dPE = ∫(mg)dh

∴ PE = mgh

Note: means "therefore"

## Relationship with kinetic energy

The kinetic or moving energy of an object is:

KE= mv2/2

where

• KE is the kinetic energy in joules (J) or foot-pounds (ft-lbs)
• v is the velocity of the object in m/s or ft/s

### Total energy

The total energy (TE) of an object related to the force of gravity is:

TE = PE + KE

This means that the initial total energy equals the final total energy:

PEi + KEi = PEf + KEf

Since PEi = mgh and PEf = 0:

mgh + mvi2/2 = mvf2/2

mgh + mvi2/2 = mvf2/2

### Falling object

In the simple case of a falling object, the initial velocity is zero (vi = 0). Thus:

mgh = mvf2/2

From this equation, you can find the velocity when the object hits the ground:

gh = vf2/2

vf2 = 2gh

vf = √2gh

### Summary

Potential energy with respect to gravity is PE = mgh. When the object is dropped, thrown downward or projected upward, its kinetic energy becomes KE = mv2/2, along with a factor of the initial velocity.

The sum of the PE and KE is the total energy, which is a constant. Equating the initial total energy with the final total energy, you can determine the final velocity of the object.

Gain confidence through small successes

## Resources and references

Ron Kurtus' Credentials

### Websites

Gravity and Potential Energy - University of Alaska

Potential Energy Calculator

Gravity Resources

### Books

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

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www.school-for-champions.com/science/
gravity_potential_energy.htm

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