Data Definitions

This section collects and defines all the data needed to build the instance models.

Center of Mass

Refname	DD:ctrOfMass
Label	Center of Mass
Symbol	${\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}$
Units	${\text{m}}$
Equation	${\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}}\,{\boldsymbol{p}\text{(}t\text{)}_{j}}}}{{m_{T}}}$
Description	${\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}$ is the Center of Mass ( ${\text{m}}$ ) ${m_{j}}$ is the mass of the j-th particle ( ${\text{kg}}$ ) ${\boldsymbol{p}\text{(}t\text{)}_{j}}$ is the position vector of the j-th particle ( ${\text{m}}$ ) ${m_{T}}$ is the total mass of the rigid body ( ${\text{kg}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy	IM:col2D and IM:transMot

Linear displacement

Refname	DD:linDisp
Label	Linear displacement
Symbol	$u\text{(}t\text{)}$
Units	${\text{m}}$
Equation	$u\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}\left(t\right)}{\,dt}$
Description	$u\text{(}t\text{)}$ is the linear displacement ( ${\text{m}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{p}\text{(}t\text{)}$ is the position ( ${\text{m}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy	IM:transMot

Linear velocity

Refname	DD:linVel
Label	Linear velocity
Symbol	$v\text{(}t\text{)}$
Units	$\frac{\text{m}}{\text{s}}$
Equation	$v\text{(}t\text{)}=\frac{\,d\boldsymbol{u}\left(t\right)}{\,dt}$
Description	$v\text{(}t\text{)}$ is the linear velocity ( $\frac{\text{m}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{u}$ is the displacement ( ${\text{m}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy	IM:transMot

Linear acceleration

Refname	DD:linAcc
Label	Linear acceleration
Symbol	$a\text{(}t\text{)}$
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	$a\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}\left(t\right)}{\,dt}$
Description	$a\text{(}t\text{)}$ is the linear acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy	IM:transMot

Angular displacement

Refname	DD:angDisp
Label	Angular displacement
Symbol	$θ$
Units	${\text{rad}}$
Equation	$θ=\frac{\,dϕ\left(t\right)}{\,dt}$
Description	$θ$ is the angular displacement ( ${\text{rad}}$ ) $t$ is the time ( ${\text{s}}$ ) $ϕ$ is the orientation ( ${\text{rad}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension).
Source	–
RefBy	IM:rotMot

Angular velocity

Refname	DD:angVel
Label	Angular velocity
Symbol	$ω$
Units	$\frac{\text{rad}}{\text{s}}$
Equation	$ω=\frac{\,dθ\left(t\right)}{\,dt}$
Description	$ω$ is the angular velocity ( $\frac{\text{rad}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) $θ$ is the angular displacement ( ${\text{rad}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension).
Source	–
RefBy	IM:rotMot

Angular acceleration

Refname	DD:angAccel
Label	Angular acceleration
Symbol	$α$
Units	$\frac{\text{rad}}{\text{s}^{2}}$
Equation	$α=\frac{\,dω\left(t\right)}{\,dt}$
Description	$α$ is the angular acceleration ( $\frac{\text{rad}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ ) $ω$ is the angular velocity ( $\frac{\text{rad}}{\text{s}}$ )
Notes	All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension).
Source	–
RefBy	IM:rotMot

Chasles’ theorem

Refname	DD:chaslesThm
Label	Chasles’ theorem
Symbol	${\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}$
Units	$\frac{\text{m}}{\text{s}}$
Equation	${\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}={\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}+ω\times{\boldsymbol{u}_{\text{O}\text{B}}}$
Description	${\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}$ is the velocity at point B ( $\frac{\text{m}}{\text{s}}$ ) ${\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}$ is the velocity at point origin ( $\frac{\text{m}}{\text{s}}$ ) $ω$ is the angular velocity ( $\frac{\text{rad}}{\text{s}}$ ) ${\boldsymbol{u}_{\text{O}\text{B}}}$ is the displacement vector between the origin and point B ( ${\text{m}}$ )
Notes	The linear velocity ${\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}$ of any point B in a rigid body is the sum of the linear velocity ${\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}$ of the rigid body at the origin (axis of rotation) and the resultant vector from the cross product of the rigid body’s angular velocity $ω$ and the displacement vector between the origin and point B ${\boldsymbol{u}_{\text{O}\text{B}}}$ . All bodies are assumed to be rigid (from A:objectTy).
Source	chaslesWiki
RefBy

Torque

Refname	DD:torque
Label	Torque
Symbol	$\boldsymbol{τ}$
Units	$\text{N}\text{m}$
Equation	$\boldsymbol{τ}=\boldsymbol{r}\times\boldsymbol{F}$
Description	$\boldsymbol{τ}$ is the torque ( $\text{N}\text{m}$ ) $\boldsymbol{r}$ is the position vector ( ${\text{m}}$ ) $\boldsymbol{F}$ is the force ( ${\text{N}}$ )
Notes	The torque on a body measures the tendency of a force to rotate the body around an axis or pivot.
Source	–
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Kinetic energy

Refname	DD:kEnergy
Label	Kinetic energy
Symbol	$KE$
Units	${\text{J}}$
Equation	$KE=m\,\frac{\|\boldsymbol{v}\text{(}t\text{)}\|^{2}}{2}$
Description	$KE$ is the kinetic energy ( ${\text{J}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ )
Notes	Kinetic energy is the measure of the energy a body possesses due to its motion. All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension). No damping occurs during the simulation (from A:dampingInvolvement).
Source	–
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Coefficient of restitution

Refname	DD:coeffRestitution
Label	Coefficient of restitution
Symbol	${C_{\text{R}}}$
Units	Unitless
Equation	${C_{\text{R}}}=-\left(\frac{{{\boldsymbol{v}\text{(}t\text{)}_{\text{f}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}{{{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}\right)$
Description	${C_{\text{R}}}$ is the coefficient of restitution (Unitless) ${{\boldsymbol{v}\text{(}t\text{)}_{\text{f}}}^{\text{A}\text{B}}}$ is the final relative velocity between rigid bodies of A and B ( $\frac{\text{m}}{\text{s}}$ ) $\boldsymbol{n}$ is the collision normal vector ( ${\text{m}}$ ) ${{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}$ is the initial relative velocity between rigid bodies of A and B ( $\frac{\text{m}}{\text{s}}$ )
Notes	The coefficient of restitution ${C_{\text{R}}}$ determines the elasticity of a collision between two rigid bodies. ${C_{\text{R}}}=1$ results in an elastic collision, ${C_{\text{R}}}\lt{}1$ results in an inelastic collision, and ${C_{\text{R}}}=0$ results in a totally inelastic collision.
Source	–
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Initial Relative Velocity Between Rigid Bodies of A and B

Refname	DD:reVeInColl
Label	Initial Relative Velocity Between Rigid Bodies of A and B
Symbol	${{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}$
Units	$\frac{\text{m}}{\text{s}}$
Equation	${{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}={\boldsymbol{v}\text{(}t\text{)}^{\text{A}\text{P}}}-{\boldsymbol{v}\text{(}t\text{)}^{\text{B}\text{P}}}$
Description	${{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}$ is the initial relative velocity between rigid bodies of A and B ( $\frac{\text{m}}{\text{s}}$ ) ${\boldsymbol{v}\text{(}t\text{)}^{\text{A}\text{P}}}$ is the velocity of the point of collision P in body A ( $\frac{\text{m}}{\text{s}}$ ) ${\boldsymbol{v}\text{(}t\text{)}^{\text{B}\text{P}}}$ is the velocity of the point of collision P in body B ( $\frac{\text{m}}{\text{s}}$ )
Notes	In a collision, the velocity of a rigid body A colliding with another rigid body B relative to that body ${{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}$ is the difference between the velocities of A and B at point P. All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy

Impulse (vector)

Refname	DD:impulseV
Label	Impulse (vector)
Symbol	$\boldsymbol{J}$
Units	$\text{N}\text{s}$
Equation	$\boldsymbol{J}=m\,Δ\boldsymbol{v}$
Description	$\boldsymbol{J}$ is the impulse (vector) ( $\text{N}\text{s}$ ) $m$ is the mass ( ${\text{kg}}$ ) $Δ\boldsymbol{v}$ is the change in velocity ( $\frac{\text{m}}{\text{s}}$ )
Notes	An impulse (vector) $\boldsymbol{J}$ occurs when a force $\boldsymbol{F}$ acts over a body over an interval of time. All bodies are assumed to be rigid (from A:objectTy).
Source	–
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Detailed derivation of impulse (vector):

Newton’s second law of motion states:

$\boldsymbol{F}=m\,\boldsymbol{a}\text{(}t\text{)}=m\,\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$

Rearranging:

$\int_{{t_{1}}}^{{t_{2}}}{\boldsymbol{F}}\,dt=m\,\left(\int_{{\boldsymbol{v}\text{(}t\text{)}_{1}}}^{{\boldsymbol{v}\text{(}t\text{)}_{2}}}{1}\,d\boldsymbol{v}\text{(}t\text{)}\right)$

Integrating the right hand side:

$\int_{{t_{1}}}^{{t_{2}}}{\boldsymbol{F}}\,dt=m\,{\boldsymbol{v}\text{(}t\text{)}_{2}}-m\,{\boldsymbol{v}\text{(}t\text{)}_{1}}=m\,Δ\boldsymbol{v}$

Potential energy

Refname	DD:potEnergy
Label	Potential energy
Symbol	$PE$
Units	${\text{J}}$
Equation	$PE=m\,\boldsymbol{g}\,h$
Description	$PE$ is the potential energy ( ${\text{J}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{g}$ is the gravitational acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $h$ is the height ( ${\text{m}}$ )
Notes	The potential energy of an object is the energy held by an object because of its position to other objects. All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension). No damping occurs during the simulation (from A:dampingInvolvement).
Source	–
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Moment of inertia

Refname	DD:momentOfInertia
Label	Moment of inertia
Symbol	$\boldsymbol{I}$
Units	$\text{kg}\text{m}^{2}$
Equation	$\boldsymbol{I}=\displaystyle\sum{{m_{j}}\,{d_{j}}^{2}}$
Description	$\boldsymbol{I}$ is the moment of inertia ( $\text{kg}\text{m}^{2}$ ) ${m_{j}}$ is the mass of the j-th particle ( ${\text{kg}}$ ) ${d_{j}}$ is the distance between the j-th particle and the axis of rotation ( ${\text{m}}$ )
Notes	The moment of inertia $\boldsymbol{I}$ of a body measures how much torque is needed for the body to achieve angular acceleration about the axis of rotation. All bodies are assumed to be rigid (from A:objectTy).
Source	–
RefBy

Software Requirements Specification for GamePhysics