In Physics

Derivatives with respect to time

In physics, we are often looking at how things change over time:

1. Velocity is the derivative of position with respect to time: $v(t) = \displaystyle{\frac{d}{dt}\big(x(t)\big)}$.
2. Acceleration is the derivative of velocity with respect to time: $\displaystyle{a(t) = \frac{d}{dt}\big(v(t)\big)= \frac{d^2 }{dt^2}}\big(x(t)\big)$.
3. Momentum (usually denoted $p$) is mass times velocity, and force ($F$) is mass times acceleration, so the derivative of momentum is $\displaystyle{\frac{dp}{dt} = \frac{d}{dt}\big(mv\big)=m \frac{dv}{dt} = ma = F}$.
   
   One of Newton's laws says that for every action there is an equal and opposite reaction, meaning that if particle 2 puts force $F$ on particle 1, then particle 1 must put force $-F$ on particle 2. But this means that the total momentum is constant, since $$\frac{d}{dt}\big(p_1 + p_2\big) = \frac{dp_1}{dt} + \frac{dp_2}{dt} = F - F = 0.$$ This is the law of conservation of momentum.

Derivatives with respect to position

In physics, we also take derivatives with respect to $x$.

1. For so-called "conservative" forces, there is a function $V(x)$ such that the force depends only on position and is minus the derivative of $V$, namely $F(x) = - \frac{dV(x)}{dx}$. The function $V(x)$ is called the potential energy. For instance, for a mass on a spring the potential energy is $\frac{1}{2}kx^2$, where $k$ is a constant, and the force is $- k x$.
2. The kinetic energy is $\frac{1}{2} m v^2$. Using the chain rule we find that the total energy is $$\frac{d }{dt} \Big(\frac{1}{2} m v^2 + V(x)\Big)= m v \frac{dv}{dt} + V'(x) \frac{dx}{dt} = m v a - F v = (m a - F) v = 0$$ since $F = m a$. This means that the total energy never changes.

These are just a few of the examples of how derivatives come up in physics. In fact, most of physics, and especially electromagnetism and quantum mechanics, is governed by differential equations in several variables. We will learn about partial derivatives in M408L/S and M408M.