Question

When a falling object reaches terminal speed, its kinetic energy reaches a constant value. However, the gravitational energy of the system consisting of object plus Earth continues to decrease. Does this violate the principle of conservation of energy? Explain why or why not.

Short answer

When a falling object reaches terminal speed, its kinetic energy reaches a constant value. However, the gravitational energy of the system consisting of object plus Earth continues to decrease. This does not violate the principle of conservation of energy as the energy is merely transferred between the object and the air.

Step-by-step solution

Definition of Kinetic Energy

The energy possessed by a body due to its motion is known as Kinetic energy. If we want something to accelerate, we must use force. It takes effort to apply force. When the job is finished, energy is transferred to the item, which moves at a new steady speed. The amount of energy transmitted is determined by the mass and speed achieved.

Explanation for the gravitational energy of the system

Terminal speed is reached when the drag force acting on the object, and the force of gravity have the same magnitude, and thus the object cannot accelerate any further (as a result of Newton's second law, since the sum of forces is zero).

It is important to remark that the air resistance force (or the drag force) is a non-conservative force. Therefore, it is expected that the object's total energy is reduced through time.

However, this does not mean that the principle of conservation of energy is violated. The energy that is 'taken' from the object goes to the surrounding air in the form of heat (due to friction between the object and the surrounding air) and kinetic energy (the surrounding air acquires speed as the object falls).

Therefore, the principle of conservation of energy is not opposed.

Calculation for the transfer of energy

The energy that the object losses is transferred to the surrounding air. Hence, energy was not created or destroyed, just transferred between the object and the surrounding air. Note that the work done by the drag force ( ${\text{W}}_{\text{NC}}$) is negative because it opposes displacement.

$\begin{array}{l}{\mathrm{E}}_{\text{ball,f}}={\mathrm{E}}_{\text{ball,i}}+{\mathrm{W}}_{\text{NC}}\\ {\mathrm{E}}_{\text{ball,i}}={\mathrm{E}}_{\text{ball,f}}-{\mathrm{W}}_{\text{NC}}\end{array}$

Substitute$-{\mathrm{W}}_{\text{NC}}={\mathrm{E}}_{\text{surroundingair}}$ into the obtained equation.

${\mathrm{E}}_{\text{ball,i}}={\mathrm{E}}_{\text{ball,f}}+{\mathrm{E}}_{\text{surroundingair}}$

Therefore, this does not oppose the principle of conservation of energy.