General Definitions

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

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:

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

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:

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

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:

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

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:

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