EQUATIONS OF MOTION IN 3D SPACE

Equations of motion in 3D space are given below for point P which is moving around the 3-dimensional space. Vector \(\vec { OP }\) is called \(\vec { { r }_{ t } }\) and subscript t indicates that vector \(\vec { { r }_{ t } }\) varies with time. The position of point P is shown with xt, yt and zt and these positions are changing with time since point P is moving around.

Vector Notation of a Particle Moving in the 3D Space

The equation of most general form of vector \(\vec { { r }_{ t } }\) is given below. Vector \(\vec { { r }_{ t } }\) is decomposed into three independent vectors and each one of these components change with time.

\(\vec { { r }_{ t } } ={ x }_{ t }\hat { x } + { y }_{ t }\hat { y } + { z }_{ t }\hat { z }\)
The velocity of this point P is the first derivative of the position wrt. time.

\(\vec { { v }_{ t } } =\frac { d\overrightarrow { { r }_{ t } } }{ dt } =\frac { dx }{ dt } \hat { x } +\frac { dy }{ dt } \hat { y } +\frac { dz }{ dt } \hat { z }\)
The notation for derivatives of x, y and z are as follows.

\(\frac { dx }{ dt } =\dot { x } , \frac { dy }{ dt } =\dot { y }, \frac { dz }{ dt } =\dot { z }\)
\(\frac { { d }^{ 2 }x }{ d{ t }^{ 2 } } =\ddot { x } ,\frac { { d }^{ 2 }y }{ d{ t }^{ 2 } } =\ddot { y } ,\frac { { d }^{ 2 }z }{ d{ t }^{ 2 } } =\ddot { z }\)
\(\vec { { v }_{ t } } =\frac { d\vec { { r }_{ t } } }{ dt } =\dot { x } \hat { x } + \dot { y } \hat { y } + \dot { z } \hat { z }\)
\(\vec { { a }_{ t } } =\frac { d\vec { { v }_{ t } } }{ dt } =\ddot { x } \hat { x } + \ddot { y } \hat { y } + \ddot { z } \hat { z }\)

The entire behaviour of the object is shown in the following figure as it moves along the x, y and z axes.

Equations of Motion in 3D-Space

In another words, 3D motion is decomposed into three 1D motions. This method is used to analyse trajectories. For example projectile motions are always analysed by decomposing projectile motion (arc) into motion along x axis and y axis.

Equations of 1 dimensional motion with constant acceleration are given below.  The first line is position as a function of time. The index t indicates that the function changes as a function of time.  The second line is velocity and it is coming from the derivative of the first function. Acceleration is the derivative of velocity function.

\({ x }_{ t }={ x }_{ 0 }+{ V }_{ { 0 }_{ x } }t+\frac { 1 }{ 2 } { a }_{ x }{ t }^{ 2 }\)
\({ { V }_{ { x }_{ t } } }={ V }_{ { 0 }_{ x } }+{ a }_{ x }t\)
\({ { a }_{ { x }_{ t } } }={ a }_{ x }\)

If there is a motion which is in 3 dimensions, the motion can be decomposed in three perpendicular axis and every x in above formulas can replaced with y which gives the entire behaviour in the y direction.  For the entire behaviour in the z axis, every x shall be replaced with z.

 

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