# Energy from Gravity of Falling Objects

by Ron Kurtus (24 May 2013)

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.

You can calculate the PE, KE and total energy (TE) for an object that is dropped. 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 for a dropped object?
• What is its kinetic energy?
• What is the total energy and final velocity for a falling object?

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

## 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.

## 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.

You can succeed

## Resources and references

Ron Kurtus' Credentials

### Websites

Gravity and Potential Energy - University of Alaska

Potential Energy Calculator

Gravity Resources

### Books

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