Central Force Problem: Conservation of Energy

The Law of Conservation of Energy states that energy can be neither created nor destroyed.

This means that the amount of energy in a closed system must always remain constant.

Conservation of energy is particularly useful for problems in which you don't know anything about the amount of time or the particular path.

For example, let us say that you would like to find the speed of an object with mass $m$ after it has fallen a height $h$, starting from rest. Call the time $t$ and the gravitational acceleration $g$.

First, we will solve the problem using a kinematics approach. Since the acceleration is constant, $$v=gt$$ $$h=\frac{1}{2}gt^2$$ Solving for $t$ in the second equation, $$t=\sqrt{\frac{2h}{g}}$$ Plugging the value of $t$ into the first equation, $$v=g\sqrt{\frac{2h}{g}}$$ $$v=\sqrt{2hg}$$ This wasn't so bad, since the acceleration is constant. However, it is not so tough to see that it could be a lot worse; for example, think about a charged particle moving on a complicated path through an electric potential.

Now, let us see the method using the conservation of energy. Computing the change in potential energy and kinetic energy, $$\Delta PE=-mgh$$ $$\Delta KE=\frac{1}{2}mv^2$$ Since the total change in energy equals zero, $$\frac{1}{2}mv^2-mgh=0$$ $$\frac{1}{2}mv^2=mgh$$ Solving for $v$ gives us: $$v=\sqrt{2hg}$$ This was a much faster and nicer way to find the speed.

From the perspective of modern physics, the Law of Conservation of Energy is no longer true, since it has been shown that mass can be converted into energy, and vice versa.

Instead, what must be used is a "combined conservation" of mass and energy, which can be derived from Albert Einstein's famous mass-energy equivalence, $E=mc^2$.