Energy from Gravity of Objects Projected Downward

by Ron Kurtus (updated 29 May 2023)

When an object at some height in projected downward and released, its initial velocity becomes a factor in its kinetic energy (KE) but does not affect its potential energy (PE).

You can calculate the PE, KE and total energy (TE) for an object that is thrown downward 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:

Is the potential energy dependent on the kinetic energy?
What is the initial and final energy for an object thrown downward?
What is the energy and final velocity for an object projected downward?

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

Potential energy of object

The potential energy (PE) of an object projected downward is independent of its velocity. It is only dependent of the height above the ground.

Initial and final PE and KE

Thus, the initial PE is:

PE_i = mgh

The final potential energy is:

PE_f = 0

The potential energy of the object is independent on its kinetic energy.

Kinetic energy when projected downward

The initial kinetic energy of an object projected downward is dependent on its initial velocity:

KE_i = mv_i²/2

where

KE_i is the initial kinetic energy in joules (J) or foot-pounds (ft-lbs)
v_i 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:

KE_f = mv_f²/2

Total energy and final velocity

The total energy relationship is:

PE_i + KE_i = PE_f + KE_f

mgh + mv_i²/2 = 0 + mv_f²/2

Divide by m, multiply by 2 and rearrange terms to get the final velocity:

v_f² = 2gh + v_i²

v_f = √(2gh + v_i²)

This equation corresponds with the equation from Velocity Equations for Objects Projected Downward:

v = √(2gy + v_i²)

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.

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