Theoretical Models

This section focuses on the general equations and laws that GamePhysics is based on.

RefnameTM:NewtonSecLawMot
LabelNewton’s second law of motion
Equation\[\boldsymbol{F}=m\,\boldsymbol{a}\text{(}t\text{)}\]
Description
  • \(\boldsymbol{F}\) is the force (\({\text{N}}\))
  • \(m\) is the mass (\({\text{kg}}\))
  • \(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes
  • The net force \(\boldsymbol{F}\) on a body is proportional to the acceleration \(\boldsymbol{a}\text{(}t\text{)}\) of the body, where \(m\) denotes the mass of the body as the constant of proportionality.
Source–
RefByIM:transMot
RefnameTM:NewtonThirdLawMot
LabelNewton’s third law of motion
Equation\[{\boldsymbol{F}_{1}}=-{\boldsymbol{F}_{2}}\]
Description
  • \({\boldsymbol{F}_{1}}\) is the force exerted by the first body (on another body) (\({\text{N}}\))
  • \({\boldsymbol{F}_{2}}\) is the force exerted by the second body (on another body) (\({\text{N}}\))
Notes
  • Every action has an equal and opposite reaction. In other words, the force \({\boldsymbol{F}_{1}}\) exerted on the second rigid body by the first is equal in magnitude and in the opposite direction to the force \({\boldsymbol{F}_{2}}\) exerted on the first rigid body by the second.
Source–
RefBy
RefnameTM:UniversalGravLaw
LabelNewton’s law of universal gravitation
Equation\[\boldsymbol{F}=G\,\frac{{m_{1}}\,{m_{2}}}{|\boldsymbol{d}|^{2}}\,\boldsymbol{\hat{d}}=G\,\frac{{m_{1}}\,{m_{2}}}{|\boldsymbol{d}|^{2}}\,\frac{\boldsymbol{d}}{|\boldsymbol{d}|}\]
Description
  • \(\boldsymbol{F}\) is the force (\({\text{N}}\))
  • \(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
  • \({m_{1}}\) is the mass of the first body (\({\text{kg}}\))
  • \({m_{2}}\) is the mass of the second body (\({\text{kg}}\))
  • \(|\boldsymbol{d}|\) is the Euclidean norm of the distance between the center of mass of two bodies (\({\text{m}}\))
  • \(\boldsymbol{\hat{d}}\) is the unit vector directed from the center of the large mass to the center of the smaller mass (\({\text{m}}\))
  • \(\boldsymbol{d}\) is the distance between the center of mass of the rigid bodies (\({\text{m}}\))
Notes
  • Two rigid bodies in the universe attract each other with a force \(\boldsymbol{F}\) that is directly proportional to the product of their masses, \({m_{1}}\) and \({m_{2}}\), and inversely proportional to the squared distance \({|\boldsymbol{d}|^{2}}\) between them.
Source–
RefByGD:accelGravity
RefnameTM:NewtonSecLawRotMot
LabelNewton’s second law for rotational motion
Equation\[\boldsymbol{τ}=\boldsymbol{I}\,α\]
Description
  • \(\boldsymbol{τ}\) is the torque (\(\text{N}\text{m}\))
  • \(\boldsymbol{I}\) is the moment of inertia (\(\text{kg}\text{m}^{2}\))
  • \(α\) is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Notes
  • The net torque \(\boldsymbol{τ}\) on a rigid body is proportional to its angular acceleration \(α\), where \(\boldsymbol{I}\) denotes the moment of inertia of the rigid body as the constant of proportionality.
  • We also assume that all rigid bodies involved are two-dimensional (from A:objectDimension).
Source–
RefByIM:rotMot