Opis ruchu w układzie kartezjańskim

A point in space has three degrees of freedom, so its position can be determined by providing three equations of motion. In rectangular coordinates, these equations will be:

\(x=f(t), y=f(t), z=f(t)\)

where the velocity components are:

\(v_x=\frac{dx}{dt}=\dot{x} \\ v_y=\frac{dy}{dt}=\dot{y} \\ v_z=\frac{dz}{dt}=\dot{z}\\\)

The total velocity of the point (always tangential to the path) is:

\(v=\sqrt{v_x^2+v_y^2+v_z^2}\)

The acceleration components of the point are:

\(a_x=\frac{dv_x}{dt}=\frac{d^2x}{dt^2}=\ddot{x} \\ a_y=\frac{dv_y}{dt}=\frac{d^2y}{dt^2}=\ddot{y} \\ a_z=\frac{dv_z}{dt}=\frac{d^2z}{dt^2}=\ddot{z} \)

The total acceleration is:

\(a=\sqrt{a_x^2+a_y^2+a_z^2}\)

The motion of the point can also be described by providing:

- the equation of the path

\(f(x,y,z)=0\)

- the equation of motion along the path

\(s=f(t)\)

In this case, the velocity of the point is given by:

\(v=\frac{ds}{dt}=\dot{s}\)

the tangential acceleration component is:

\(a_t=\frac{dv}{dt}=\frac{d^2s}{dt^2}=\ddot{s}\)

and the normal component is:

\(a_n=\frac{v^2}{\rho}\)

where \(\rho\) is the radius of curvature of the path.

The total acceleration of the point is determined by the formula:

\(a=\sqrt{a_t^2+a_n^2}\)