Acceleration as a Derivative
We usually see Newton's Second Law as \(F = ma\). But in calculus, we write it as a 2nd-order differential equation:
\[F(x, t) = m \frac{d^2 x}{dt^2}\]
Solving for the motion of a particle means solving this DE for the position function \(x(t)\).
Worked Examples
Example 1: Gravity
For a falling object, \(F = -mg\).
- DE: \(m x'' = -mg \implies x'' = -g\).
- Integrate once: \(x' = -gt + v_0\).
- Integrate again: \(x = -\frac{1}{2}gt^2 + v_0 t + x_0\).
- Result: The standard kinematic equation for falling bodies.
The Bridge to Quantum Mechanics
Classical mechanics predicts a precise path \(x(t)\) based on Newton's Law. Quantum Mechanics replaces this with the Ehrenfest Theorem, which says that the average position of a quantum wavepacket follows Newton's Law: \(m \frac{d^2 \langle x \rangle}{dt^2} = \langle -\nabla V \rangle\). This means that for big objects, the quantum randomness averages out, and the "most probable" path is exactly the one Newton predicted. Quantum Mechanics doesn't break Newton's Laws; it "explains" them as the limit of wave behavior.