Basic mechanics

Konrad Hinsen

Context kinematics
uses quantities/length
uses quantities/time
uses quantities/velocity
uses quantities/acceleration

1 Kinematics

The derived quantities V and A are obtained as quotients of the fundamental quantities L and T.

Velocities are obtained by dividing a length by a time:

L ÷ Tnz : V
Lnz ÷ Tnz : Vnz

Accelerations are the result of dividing a velocity by a time:

V ÷ Tnz : A
Vnz ÷ Tnz : Anz

Context kinematics-example
extends kinematics

1.1 Example

We consider a point that moves on a straight line starting at time 0 from the origin. At time t1 : Tnz it has distance d1 : L from the origin, at time t2 : Tnz, t2 > t1 the distance is d2 : L.

The average velocity from time 0 to t1 is then

v1 : V
v1 ⇒ d1 ÷ t1.

and the average velocity between 0 and t2 is

v2 : V
v2 ⇒ d2 ÷ t2.

The average acceleration is given by

a : A
a ⇒ 2 × ((v2 - v1) ÷ (t2 - t1))
a ⇒ 2 × (((d2 ÷ t2) - (d1 ÷ t1)) ÷ (t2 - t1)).

Context kinematics-nummerical-example
extends kinematics-example

Introducing a time unit s : Tnz and a length unit m : Lnz, we can assign numerical values:

t1 ⇒ 3 × s, d1 ⇒ 20 × m
t2 ⇒ 6 × s, d2 ⇒ 50 × m
v1 ⇒ 20/3 × (m ÷ s)
v2 ⇒ 25/3 × (m ÷ s)
a ⇒ 10/9 × (m ÷ s ÷ s).

Context time-dependent-kinematics
extends kinematics
includes transformed quantities/function-with-finite-difference-template
includes transformed quantities/function-with-finite-difference-template

2 Time-dependent kinematics

When describing motion, quantities L, V, and A become functions of T. These time-dependent quantities are written as T→L ⊆ T→Q, T→V ⊆ T→Q, and T→A ⊆ T→Q, with each one being the time derivative of its predecessor. The sort T→Q ⊆ Q→Q covers all these time-dependent quantities.

Context dynamics
extends time-dependent-kinematics
uses quantities/mass
uses quantities/force
includes transformed quantities/function-template

3 Dynamics

Extending kinematics to dynamics requires M for masses, F for forces, and T→F ⊆ T→Q for time-dependent forces, plus the following relations between these quantities:

M × A : F
Mnz × Anz : Fnz
M × T→A : T→F
(m × f)[t] ⇒ m × f[t]
  ∀ m : M
  ∀ t : T
  ∀ f : T→A