SfC Home > Physical Science > Physics > Gravity >
Explanation of Potential Energy of Gravity by Ron Kurtus - Succeed in Understanding Physics. Also refer to PE, kinetic energy, KE, initial velocity, falling objects, physical science, School for Champions. Copyright © Restrictions
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, in that position, 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.
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 energy and final velocity for a dropped object?
- What is the energy and final velocity for an object thrown downward?
- What is the energy and final velocity for an object projected upward?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Energy for falling objects
An object held at a given height above the ground has an initial potential energy. When it is dropped, it gains kinetic energy. The total energy of the object is then used to find the velocity when the object hits the ground.
Potential energy for falling object
The equation for the object's initial PE with respect to gravity is::
PEi = mgh
where
- PEi is the initial 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.
When the object reaches the ground, h = 0 and thus the final potential energy is:
PEf = 0
Note: In reality, there is still a gravitational force on the object at the surface of the Earth, so the object has a gravitational potential energy at that point. But since the object cannot go anywhere, we say its PE from gravity is zero.
Kinetic energy for falling object
Kinetic energy (KE) is the energy of motion. Since the object is not moving at the initial position, the initial KE is:
KEi = 0
Once the object is released, it accelerates downward. When the object reaches the ground, its kinetic energy is:
KEf = mvf2/2
where
- KEf is the kinetic energy at the ground in joules (J) or foot-pounds (ft-lbs)
- vf is the downward velocity of the object at the ground in m/s or ft/s
Total energy for falling object
The total energy of the object is:
TE = PE + KE
The total energy is a constant value, provided no external forces besides gravity act on the object. Thus, the initial total energy equals the final total energy:
TEi = TEf
PEi + KEi = PEf + KEf
When the object is simply dropped,
mgh + 0 = 0 + mvf2/2
mgh = mvf2/2
Final velocity for falling object
From that equivalence, you can determine the final velocity of the dropped object. Divide by m and multiply by 2:
vf2 = 2gh
vf = √(2gh)
This is equivalent to v = √(2gy) that is given in Velocity Equations for Falling Objects.
Energy for objects projected downward
When an object at some height in projected downward and released, its initial velocity becomes a factor in the KE. However, it does not affect the PE.
Initial and final PE and KE
Potential energy when projected downward
The PE of an object is independent of its velocity. It is only dependent of the height above the ground. Thus, the initial PE is:
PEi = mgh
The final potential energy is:
PEf = 0
Kinetic energy when projected downward
The initial kinetic energy of an object projected downward is dependent on its initial velocity:
KEi = mvi2/2
where
- KEi is the initial kinetic energy in joules (J) or foot-pounds (ft-lbs)
- vi is the initial velocity of the object in m/s or ft/s
The object accelerates until it hits the ground at a final KE:
KEf = mvf2/2
Total energy and final velocity
The total energy relationship is:
PEi + KEi = PEf + KEf
mgh + mvi2/2 = 0 + mvf2/2
Divide by m, multiply by 2 and rearrange terms to get the final velocity:
vf2 = 2gh + vi2
vf = √(2gh + vi2)
This equation corresponds with the equation from Velocity Equations for Objects Projected Downward:
v = √(2gy + vi2)
where y is the displacement below the starting point.
Energy for objects projected upward
When an object is projected upward from a given height, it travels until it reaches a maximum displacement, at which time its velocity is zero. The object then falls to the ground from that displacement.
Potential energy when projected upward
The initial potential energy of an object projected upward is:
PEi = mgh
As the object is moving upwards, the PE increases according to its displacement. The maximum displacement is:
ym = −vi2/2g
where ym is the maximum displacement from the starting point.
(See Displacement Equations for Objects Projected Upward for more information.)
Note: The value of ym is a negative number, because the motion is in the opposite direction of gravity.
However, the object is projected upward from a given height. Thus, the maximum height above the ground (hm) that it reaches is:
hm = h − ym
The equation for the PE at the maximum height is then:
PEm = mghm
or
PEm = mgh − mgym
Substituting for ym:
PEm = mgh + mvi2/2
Considerations concerning PE can be made from the initial height.
Kinetic energy when projected upward
The initial KE is:
KEi = mvi2/2
At the maximum displacement, v = 0 and thus:
KEm = 0
When the object falls and finally reaches the ground:
KEf = mvf2/2
Total energy and final velocity
To determine the final velocity, consider the total energy at the maximum displacement and compare it with the total energy at the ground:
PEm + KEm = PEf + KEf
(mgh + mvi2/2) + 0 = 0 + mvf2/2
Divide by m, multiply by 2 and rearrange terms to get the final velocity:
vf2 = 2gh + vi2
vf = √(2gh + vi2)
This compares with the equation in Velocity Equations for Objects Projected Upward:
v = √(2gy + vi2)
where y is the displacement below the starting point.
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
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
What do you think?
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.
Share link
Click on a button to share the link for this page:
Or use our form to send this link to yourself or a friend.
Students and researchers
The Web address of this page is:
www.school-for-champions.com/science/gravity_potential_energy.htm
Please include it as a link on your website or as a reference in your report, document, or thesis.
Where are you now?
Potential Energy of Gravity