General Definitions
This section collects the laws and equations that will be used to build the instance models.
Refname | GD:rectVel |
---|---|
Label | Rectilinear (1D) velocity as a function of time for constant acceleration |
Units | ms |
Equation | v(t)=vi+act |
Description |
|
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 ac. The initial velocity (at t=0, from A:timeStartZero) is represented by vi. From TM:acceleration in 1D, and using the above symbols we have:
ac=dvdt
Rearranging and integrating, we have:
∫vvi1dv=∫t0acdt
Performing the integration, we have the required equation:
v(t)=vi+act
Refname | GD:rectPos |
---|---|
Label | Rectilinear (1D) position as a function of time for constant acceleration |
Units | m |
Equation | p(t)=pi+vit+act22 |
Description |
|
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 pi. From TM:velocity in 1D, and using the above symbols we have:
v=dpdt
Rearranging and integrating, we have:
∫ppi1dp=∫t0vdt
From GD:rectVel, we can replace v:
∫ppi1dp=∫t0vi+actdt
Performing the integration, we have the required equation:
p(t)=pi+vit+act22
Refname | GD:velVec |
---|---|
Label | Velocity vector as a function of time for 2D motion under constant acceleration |
Units | ms |
Equation | v(t)=[vxi+axctvyi+ayct] |
Description |
|
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 v(t)=[vxvy] and the acceleration vector as a(t)=[axay]. The acceleration is assumed to be constant (A:constAccel) and the constant acceleration vector is represented as ac=[axcayc]. The initial velocity (at t=0, from A:timeStartZero) is represented by vi=[vxivyi]. Since we have a Cartesian coordinate system, GD:rectVel can be applied to each coordinate of the velocity vector to yield the required equation:
v(t)=[vxi+axctvyi+ayct]
Refname | GD:posVec |
---|---|
Label | Position vector as a function of time for 2D motion under constant acceleration |
Units | m |
Equation | p(t)=[pxi+vxit+axct22pyi+vyit+ayct22] |
Description |
|
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 p(t)=[pxpy], the velocity vector as v(t)=[vxvy], and the acceleration vector as a(t)=[axay]. The acceleration is assumed to be constant (A:constAccel) and the constant acceleration vector is represented as ac=[axcayc]. The initial velocity (at t=0, from A:timeStartZero) is represented by vi=[vxivyi]. Since we have a Cartesian coordinate system, GD:rectPos can be applied to each coordinate of the position vector to yield the required equation:
p(t)=[pxi+vxit+axct22pyi+vyit+ayct22]