# 4th Order Point Mass (Longitudinal)

Calculate fourth-order point mass

## Library

Equations of Motion/Point Mass

## Description

The 4th Order Point Mass (Longitudinal) block performs the calculations for the translational motion of a single point mass or multiple point masses.

The translational motions of the point mass [XEast XUp]T are functions of airspeed (V) and flight path angle (γ),

`$\begin{array}{c}{F}_{x}=m\stackrel{˙}{V}\\ {F}_{z}=mV\stackrel{˙}{\gamma }\\ {\stackrel{˙}{X}}_{East}=V\mathrm{cos}\gamma \\ {\stackrel{˙}{X}}_{Up}=V\mathrm{sin}\gamma \end{array}$`

where the applied forces [Fx Fz]T are in a system defined as follows: x-axis is in the direction of vehicle velocity relative to air, z-axis is upward, and y-axis completes the right-handed frame. The mass of the body m is assumed constant.

## Parameters

Units

Specifies the input and output units:

Units

Forces

Velocity

Position

`Metric (MKS)`

Newton

Meters per second

Meters

`English (Velocity in ft/s)`

Pound

Feet per second

Feet

`English (Velocity in kts)`

Pound

Knots

Feet

Initial flight path angle

The scalar or vector containing the initial flight path angle of the point mass(es).

Initial airspeed

The scalar or vector containing the initial airspeed of the point mass(es).

Initial downrange

The scalar or vector containing the initial downrange of the point mass(es).

Initial altitude

The scalar or vector containing the initial altitude of the point mass(es).

Initial mass

The scalar or vector containing the mass of the point mass(es).

## Inputs and Outputs

InputDimension TypeDescription
First Contains the force in x-axis in selected units.
Second Contains the force in z-axis in selected units.
OutputDimension TypeDescription
First Contains the flight path angle in radians.
Second Contains the airspeed in selected units.
Third Contains the downrange or amount traveled East in selected units.
Fourth Contains the altitude or amount traveled Up in selected units.

## Assumptions and Limitations

The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the “fixed stars” to be neglected.