Physics

To solve this problem, we will use the concepts of mechanical energy conservation and the relationship between kinetic and potential energy.

 

Define the energies at height

The total mechanical energy of the object is conserved, so the sum of the object's kinetic energy and potential energy remains constant. At any height , we have:

- Kinetic energy:

- Gravitational potential energy:

 

The total mechanical energy at launch is entirely kinetic, since the object starts from the ground ():

 

At height , the object has both potential energy and kinetic energy. We are told that the gravitational potential energy is three times the kinetic energy, so:

Substitute the expressions for and :

 

Relate the velocities and energies

At height , the total energy is still . The mechanical energy at height is the sum of the potential energy and kinetic energy:

 

Using conservation of mechanical energy: