Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Kinematic Equations

A set of equations relating displacement, velocity, acceleration, and time for constant acceleration motion: v = v₀ + at, x = v₀t + rac{1}{2}at², x = v_t.
📜

The statement of the theorem

Let x(t)∈Rn\mathbf{x}(t) \in \mathbb{R}^n be the position vector of a particle, v(t)=dxdt\mathbf{v}(t) = \frac{d\mathbf{x}}{dt} its velocity vector, and a(t)=dvdt\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} its acceleration vector. Assume the acceleration is constant, i.e., a(t)=a0∈Rn\mathbf{a}(t) = \mathbf{a}_0 \in \mathbb{R}^n. Then, the following relationships hold for the initial conditions x(0)=x0\mathbf{x}(0) = \mathbf{x}_0 and v(0)=v0\mathbf{v}(0) = \mathbf{v}_0: \begin{enumerate} \item \textbf{Velocity:} v(t)=v0+∫0ta0τdτ=v0+a0t\mathbf{v}(t) = \mathbf{v}_0 + \int_{0}^{t} \mathbf{a}_0 \tau d\tau = \mathbf{v}_0 + \mathbf{a}_0 t. \item \textbf{Position:} x(t)=x0+∫0tv(τ)dτ=x0+v0t+12a0t2\mathbf{x}(t) = \mathbf{x}_0 + \int_{0}^{t} \mathbf{v}(\tau) d\tau = \mathbf{x}_0 + \mathbf{v}_0 t + \frac{1}{2} \mathbf{a}_0 t^2. \item \textbf{Displacement (Time-independent):} x(t)−x0=x0+v0t+12a0t2−x0= v0t+12a0t2\mathbf{x}(t) - \mathbf{x}_0 = \mathbf{x}_0 + \mathbf{v}_0 t + \frac{1}{2} \mathbf{a}_0 t^2 - \mathbf{x}_0 = \ \mathbf{v}_0 t + \frac{1}{2} \mathbf{a}_0 t^2. (This is derived by eliminating tt between the first two equations, yielding the scalar form vf2=vi2+2a(xf−xi)v_f^2 = v_i^2 + 2a(x_f - x_i) when considering the magnitude squared, ∣v(t)−v0∣2=∣a0t∣2|\mathbf{v}(t) - \mathbf{v}_0|^2 = |\mathbf{a}_0 t|^2).
Source: Wikipedia