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Acceleration

Acceleration is the rate of change of velocity with respect to time, representing the object's change in speed or direction.
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The statement of the theorem

Let x:I→R3\mathbf{x}: I \to \mathbb{R}^3 be a differentiable position vector, where I=[t0,tf]⊂RI = [t_0, t_f] \subset \mathbb{R} is the time interval. Define the velocity vector v(t)\mathbf{v}(t) as the first derivative of x(t)\mathbf{x}(t) with respect to time tt: v(t)=dxdt\mathbf{v}(t) = \frac{d\mathbf{x}}{dt}. The acceleration vector a(t)\mathbf{a}(t) is then defined as the time derivative of the velocity vector: a(t)=dvdt=d2xdt2.\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}. Furthermore, if a\mathbf{a} is expressed in Cartesian coordinates, a(t)=(ax(t),ay(t),az(t))\mathbf{a}(t) = (a_x(t), a_y(t), a_z(t)), where ai(t)=d2xidt2a_i(t) = \frac{d^2x_i}{dt^2} for i∈{x,y,z}i \in \{x, y, z\}.