Description of motion in the Cartesian system

Motion of a point in rectangular coordinates

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

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

a) Components of velocity

From where the components of velocity:

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

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

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

b) Components of acceleration

Components of acceleration of the point:

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

Total acceleration:

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

Description of motion through the equation of the path

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

• the equation of the path: \(f(x,y,z)=0\)
• the equation of motion along the path: \(s=f(t)\)

a) Velocity of the point

The velocity of the point is then:

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

b) Components of acceleration

Tangential component of acceleration:

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

Normal component:

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

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

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

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