Energy from Gravity for Objects Projected Upward

by Ron Kurtus (updated 29 May 2023)

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.

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

Potential energy when projected upward

The initial potential energy of an object projected upward is:

PE_i = mgh

As the object is moving upwards, the PE increases according to its displacement. The maximum displacement is:

y_m = −v_i²/2g

where y_m is the maximum displacement from the starting point.

(See Displacement Equations for Objects Projected Upward for more information.)

Note: The value of y_m 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 (h_m) that it reaches is:

h_m = h − y_m

The equation for the PE at the maximum height is then:

PE_m = mgh_m

PE_m = mgh − mgy_m

Substituting for y_m:

PE_m = mgh + mv_i²/2

Considerations concerning PE can be made from the initial height.

Kinetic energy when projected upward

The initial KE is:

KE_i = mv_i²/2

At the maximum displacement, v = 0 and thus:

KE_m = 0

When the object falls and finally reaches the ground:

KE_f = mv_f²/2

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:

PE_m + KE_m = PE_f + KE_f

(mgh + mv_i²/2) + 0 = 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 compares with the equation in Velocity Equations for Objects Projected Upward:

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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