General Definitions

This section collects the laws and equations that will be used to build the instance models.

Rectilinear (1D) velocity as a function of time for constant acceleration

Refname	GD:rectVel
Label	Rectilinear (1D) velocity as a function of time for constant acceleration
Units	$\frac{\text{m}}{\text{s}}$
Equation	$v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t$
Description	$v\text{(}t\text{)}$ is the 1D speed ( $\frac{\text{m}}{\text{s}}$ ) ${v^{\text{i}}}$ is the initial speed ( $\frac{\text{m}}{\text{s}}$ ) ${a^{c}}$ is the constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ )
Source	hibbeler2004 (pg. 8)
RefBy	GD:velVec and GD:rectPos

Detailed derivation of rectilinear velocity:

Assume we have rectilinear motion of a particle (of negligible size and shape, from A:pointMass); that is, motion in a straight line. The velocity is $v$ and the acceleration is $a$ . The motion in TM:acceleration is now one-dimensional with a constant acceleration, represented by ${a^{c}}$ . The initial velocity (at $t=0$ , from A:timeStartZero) is represented by ${v^{\text{i}}}$ . From TM:acceleration in 1D, and using the above symbols we have:

${a^{c}}=\frac{\,dv}{\,dt}$

Rearranging and integrating, we have:

$\int_{{v^{\text{i}}}}^{v}{1}\,dv=\int_{0}^{t}{{a^{c}}}\,dt$

Performing the integration, we have the required equation:

$v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t$

Rectilinear (1D) position as a function of time for constant acceleration

Refname	GD:rectPos
Label	Rectilinear (1D) position as a function of time for constant acceleration
Units	${\text{m}}$
Equation	$p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2}$
Description	$p\text{(}t\text{)}$ is the 1D position ( ${\text{m}}$ ) ${p^{\text{i}}}$ is the initial position ( ${\text{m}}$ ) ${v^{\text{i}}}$ is the initial speed ( $\frac{\text{m}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) ${a^{c}}$ is the constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	hibbeler2004 (pg. 8)
RefBy	GD:posVec

Detailed derivation of rectilinear position:

Assume we have rectilinear motion of a particle (of negligible size and shape, from A:pointMass); that is, motion in a straight line. The position is $p$ and the velocity is $v$ . The motion in TM:velocity is now one-dimensional. The initial position (at $t=0$ , from A:timeStartZero) is represented by ${p^{\text{i}}}$ . From TM:velocity in 1D, and using the above symbols we have:

$v=\frac{\,dp}{\,dt}$

Rearranging and integrating, we have:

$\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{v}\,dt$

From GD:rectVel, we can replace $v$ :

$\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{{v^{\text{i}}}+{a^{c}}\,t}\,dt$

Performing the integration, we have the required equation:

$p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2}$

Velocity vector as a function of time for 2D motion under constant acceleration

Refname	GD:velVec
Label	Velocity vector as a function of time for 2D motion under constant acceleration
Units	$\frac{\text{m}}{\text{s}}$
Equation	$\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\{{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t\end{bmatrix}$
Description	$\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ ) ${{v_{\text{x}}}^{\text{i}}}$ is the $x$ -component of initial velocity ( $\frac{\text{m}}{\text{s}}$ ) ${{a_{\text{x}}}^{\text{c}}}$ is the $x$ -component of constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ ) ${{v_{\text{y}}}^{\text{i}}}$ is the $y$ -component of initial velocity ( $\frac{\text{m}}{\text{s}}$ ) ${{a_{\text{y}}}^{\text{c}}}$ is the $y$ -component of constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	–
RefBy

Detailed derivation of velocity vector:

For a two-dimensional Cartesian coordinate system (A:twoDMotion and A:cartSyst), we can represent the velocity vector as $\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{v_{\text{x}}}\\{v_{\text{y}}}\end{bmatrix}$ and the acceleration vector as $\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}{a_{\text{x}}}\\{a_{\text{y}}}\end{bmatrix}$ . The acceleration is assumed to be constant (A:constAccel) and the constant acceleration vector is represented as ${\boldsymbol{a}^{\text{c}}}=\begin{bmatrix}{{a_{\text{x}}}^{\text{c}}}\\{{a_{\text{y}}}^{\text{c}}}\end{bmatrix}$ . The initial velocity (at $t=0$ , from A:timeStartZero) is represented by ${\boldsymbol{v}^{\text{i}}}=\begin{bmatrix}{{v_{\text{x}}}^{\text{i}}}\\{{v_{\text{y}}}^{\text{i}}}\end{bmatrix}$ . Since we have a Cartesian coordinate system, GD:rectVel can be applied to each coordinate of the velocity vector to yield the required equation:

$\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\{{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t\end{bmatrix}$

Position vector as a function of time for 2D motion under constant acceleration

Refname	GD:posVec
Label	Position vector as a function of time for 2D motion under constant acceleration
Units	${\text{m}}$
Equation	$\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\{{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2}\end{bmatrix}$
Description	$\boldsymbol{p}\text{(}t\text{)}$ is the position ( ${\text{m}}$ ) ${{p_{\text{x}}}^{\text{i}}}$ is the $x$ -component of initial position ( ${\text{m}}$ ) ${{v_{\text{x}}}^{\text{i}}}$ is the $x$ -component of initial velocity ( $\frac{\text{m}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) ${{a_{\text{x}}}^{\text{c}}}$ is the $x$ -component of constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${{p_{\text{y}}}^{\text{i}}}$ is the $y$ -component of initial position ( ${\text{m}}$ ) ${{v_{\text{y}}}^{\text{i}}}$ is the $y$ -component of initial velocity ( $\frac{\text{m}}{\text{s}}$ ) ${{a_{\text{y}}}^{\text{c}}}$ is the $y$ -component of constant acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfLandingDist and IM:calOfLandingTime

Detailed derivation of position vector:

For a two-dimensional Cartesian coordinate system (A:twoDMotion and A:cartSyst), we can represent the position vector as $\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{p_{\text{x}}}\\{p_{\text{y}}}\end{bmatrix}$ , the velocity vector as $\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{v_{\text{x}}}\\{v_{\text{y}}}\end{bmatrix}$ , and the acceleration vector as $\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}{a_{\text{x}}}\\{a_{\text{y}}}\end{bmatrix}$ . The acceleration is assumed to be constant (A:constAccel) and the constant acceleration vector is represented as ${\boldsymbol{a}^{\text{c}}}=\begin{bmatrix}{{a_{\text{x}}}^{\text{c}}}\\{{a_{\text{y}}}^{\text{c}}}\end{bmatrix}$ . The initial velocity (at $t=0$ , from A:timeStartZero) is represented by ${\boldsymbol{v}^{\text{i}}}=\begin{bmatrix}{{v_{\text{x}}}^{\text{i}}}\\{{v_{\text{y}}}^{\text{i}}}\end{bmatrix}$ . Since we have a Cartesian coordinate system, GD:rectPos can be applied to each coordinate of the position vector to yield the required equation:

$\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\{{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2}\end{bmatrix}$

Software Requirements Specification for Projectile