Energy Conservation

The energy in a closed system is always conserved. It cannot be generated or destroyed. As an example, we can take a look at a car that moves at a certain speed. When the driver removes his foot from the gas pedal, the car starts to get slower, mainly because of friction and air resistance. For the car itself, the energy is not conserved. However, we have to take the total system into account which contains all parts of the car, the street, and the air. In all places, where friction occurs, e.g. between the wheels and the street, heat is created, because the kinetic energy of the car is transformed into the motion energy of the atoms and molecules of the involved parts. Similarly, the air heats slightly up due to collisions of the gas molecules with the moving car. Another easy example is the one of a falling object from the height $h$ to the ground. In the beginning, it has the potential energy $E_\mathrm{pot}$ and the kinetic energy $E_\mathrm{kin}$. After touching the ground these values change to $E'_\mathrm{pot}$ and $E'_\mathrm{kin}$. The law of energy conservation states now that the total energy is constant at any given point in time. We can therefore write $$E_\mathrm{pot} + E_\mathrm{kin} = E'_\mathrm{pot} + E'_\mathrm{kin}$$ We now assume that the object was in rest before falling down. The kinetic energy, in the beginning, is there 0. Additionally, we define the potential energy of the object at ground level as 0. The above-mentioned equation then simplifies to: $$mgh = \frac{1}{2}mv^2$$ In order to calculate the final speed of the object, this equation can be solved for \(v\) which results in $$v=\sqrt{2gh}$$ since $g$ cancels each other out on both sides. The obtained result is totally equal to the one that we have derived in the kinematics section for a free-falling object. The main difference, however, is that we have now shown that this formula can be applied to all trajectories in the gravitational field near the surface of the earth.

• The Falling Flowerpot (+)