In a closed system with no external forces acting on it, the total mechanical energy remains constant. This means that the sum of potential energy (PE) and kinetic energy (KE) stays the same.

Initially, the total mechanical energy of the object is the sum of its potential and kinetic energies: $E_{\text{total}} = PE_i + KE_i = 10 \, \text{J} + 20 \, \text{J} = 30 \, \text{J}$

When the potential energy increases to $PE_f = 40 \, \text{J}$, the total mechanical energy still must remain 30 J. Thus, the new kinetic energy can be found by subtracting the new potential energy from the total energy: $KE_f = E_{\text{total}} - PE_f = 30 \, \text{J} - 40 \, \text{J} = -10 \, \text{J}$

Since kinetic energy cannot be negative, this suggests that the given conditions (PEf > PEi) are not physically possible in a closed system without external energy input. Therefore, the object cannot have potential energy of 40 J if the initial total energy is 30 J.

Would you like more details on energy conservation, or have any questions on this?

Here are 5 related questions:

1. What happens when an external force acts on the object?
2. How can mechanical energy be converted into heat or other forms of energy?
3. What is the relationship between potential energy and height in a gravitational field?
4. How do we account for energy losses in real-world systems?
5. Can mechanical energy change in an open system?

Tip: Always check whether total mechanical energy remains constant before solving for individual energies!