How to Calculate the potential energy of a falling object?

Answer 1

Gravitational potential energy (GPE) is essentially a measure of stored energy. It is defined as being a function of gravity (9.8m/s2 on Earth), mass and perpedicular distance above the ground.

Since work is a measure of energy the GPE is a measure of potential work. Work is defined as:

Work = Force * Distance = Fd

Newton's laws of motion are then used to replace force with some function of mass and acceleration due to gravity:

Force = Mass * Acceleration = ma

or, in this case,

Force = Mass * Gravity = mg

Therefore:

GPE = Mass * Gravity * Distance = mgd

For a falling object it is losing potential energy as it moves closer to the ground (because the distance value in the equation above is reducing) so the distance between the object and the ground defines the remaining GPE of the falling object for the remainder of the fall after this point. This equation will only tell you the GPE relative to distance from the ground, as mass and gravity are constants.

If you wanted to take that a bit further you can factor in the speed of falling and derive equations to calculate the GPE at a specific time interval relative to when the object is released. To do this we need Newton's equations of motion (I've cancelled out and modified the below equation for simplicity):

Distance = ( Gravity * Time * Time ) / 2 = 0.5gt2

Plugging this into the GPE equation we get,

Remaining GPE = mg(d-0.5gt2)

Examples:

So if an object with a mass of 100kg is held at 50m above the ground on Earth it will have a GPE of:

GPE = mgd = 100 * 9.8 * 50 = 49.00kJ

If the object is released and we want to know the remaining GPE after 1 second:

Remaining GPE = mg(d-0.5gt2) = 100 * 9.8 * ( 50 - ( 0.5 * 9.8 * 12 ) ) = 44.20kJ

Remaining GPE after 2 seconds:

Remaining GPE = mg(d-0.5gt2) = 100 * 9.8 * ( 50 - ( 0.5 * 9.8 * 22 ) ) = 29.79kJ