Newton's Laws & Projectile Motion
Forces, momentum, and the canonical 2D projectile problem.
Newton's three laws:
- A body at rest or in uniform motion remains so unless acted on by a net force.
- $\mathbf F = d\mathbf p/dt$; for constant mass, $\mathbf F = m\mathbf a$.
- Forces come in equal-and-opposite pairs.
For a projectile launched at speed $v_0$ and angle $\theta$ in uniform gravity $\mathbf g = -g\hat y$, horizontal and vertical motions decouple:
$$x(t) = v_0\cos\theta\, t, \quad y(t) = v_0\sin\theta\, t - \tfrac{1}{2}g t^2.$$Range $R = v_0^2 \sin 2\theta / g$ is maximized at $\theta = 45°$; the trajectory is a parabola.