Vector kinematics

Содержание и основные понятия

The position of the material point sets the radius-vector, the change is displacement. The instantaneous velocity is directed tangential to the trajectory, the acceleration describes the change of the velocity vector.
GEF · 10 grade
PIF · Baseline
Interactive model
radius-vector
instantaneous speed
Tangential acceleration
normal acceleration
01
The physical picture
What happens and what boundaries the model has.
Main message
The position of the material point sets the radius-vector, the change is displacement. The instantaneous velocity is directed tangential to the trajectory, the acceleration describes the change of the velocity vector.
The average and instantaneous speeds differ in the limit transition to a short time interval.
The acceleration may change the speed module, direction or both at the same time.
The independent projection of the vector equation allows the spatial task to be solved by coordinates.
How to reason
First, name the system and the interaction, then select the values, and only then write down the law. The formula works within the model's assumptions.
02
Laws and values
Symbols should be read as a relationship of values, not as a recipe for substitution.
v
⃗
ср
=
Δ
r
⃗
Δ
t
\vec v_{\mathrm{ср}}=\frac{\Delta\vec r}{\Delta t}
v
cf
​
=
Δ
t
Δ
r
​
Average speed
a
⃗
ср
=
Δ
v
⃗
Δ
t
\vec a_{\mathrm{ср}}=\frac{\Delta\vec v}{\Delta t}
a
cf
​
=
Δ
t
Δ
v
​
Average acceleration
Review of the situation
The speed module is constant in even motion on the circle, but acceleration is not equal to zero due to the rotation of the velocity vector.
04
Links and application
In the real world
The speed module is constant in even motion on the circle, but acceleration is not equal to zero due to the rotation of the velocity vector.
Model verification
Compare the prediction with observation, evaluate the units, the order of magnitude and the influence of the factors not taken into account. Disparity does not always mean an error of law: often idealization is no longer sufficient.
C
What needs to be understood
Outcome 1
Explain the physical meaning of the topic "Vocular kinematics" through the model, not just through the formula.
Outcome 2
Clear and correctly use terms: radius vector, instantaneous velocity, tangential acceleration.
Outcome 3
Indicate the conditions for the applicability of the law, verify units and limit cases.
T
Terms of the topic
The definition should indicate the physical meaning, not a substitute for a domestic synonym.
radius-vector
A vector from the beginning of the coordinates to the position of the material point.
instantaneous speed
Average speed limit when the time interval is zeroed; directed at the tangent to the trajectory.
Tangential acceleration
Composing accelerations along the tangent to the trajectory, changing the speed module.
normal acceleration
Composing accelerations, directed to the centre of curvature and changing direction of speed.
F
Origin and conclusion of the formulas
The fundamental law cannot be "proven" by algebras: for each record, it is stated whether it is a definition, an experimental law, or a model effect.
v
⃗
ср
=
Δ
r
⃗
Δ
t
\vec v_{\mathrm{ср}}=\frac{\Delta\vec r}{\Delta t}
v
cf
​
=
Δ
t
Δ
r
​
definition
Average speed.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
Δ
r
⃗
=
r
⃗
2
−
r
⃗
1
\Delta\vec r=\vec r_2-\vec r_1
Δ
r
=
r
2
​
−
r
1
​
Moving is equal to changing the radius vector over the selected interval.
v
⃗
ср
≡
Δ
r
⃗
Δ
t
\vec v_{\mathrm{ср}}\equiv\frac{\Delta\vec r}{\Delta t}
v
cf
​
≡
Δ
t
Δ
r
​
Average speed is defined as movement per unit of time; it is a definition, not a withdrawal from the law of motion.
Conditions:
For the final interval Δt > 0 in the selected system, the result depends on the system of the countdown and in general is not equal to the average travel speed.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
a
⃗
ср
=
Δ
v
⃗
Δ
t
\vec a_{\mathrm{ср}}=\frac{\Delta\vec v}{\Delta t}
a
cf
​
=
Δ
t
Δ
v
​
definition
Average acceleration.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
Δ
v
⃗
=
v
⃗
2
−
v
⃗
1
\Delta\vec v=\vec v_2-\vec v_1
Δ
v
=
v
2
​
−
v
1
​
First, they find vector speed change.
a
⃗
ср
≡
Δ
v
⃗
Δ
t
\vec a_{\mathrm{ср}}\equiv\frac{\Delta\vec v}{\Delta t}
a
cf
​
≡
Δ
t
Δ
v
​
The average acceleration by definition shows a change in speed per unit time.
Conditions:
For the final interval Δt > 0 in an inertial or clearly defined system, the acceleration variable describes the entire time rather than a single moment.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
!
Model boundaries and typical errors
Error: Forget model condition
The formula cannot be moved to any situation simply because of the overlap of letters. The key condition of this subject is that the average and instantaneous speeds differ in the limit of the transition to a short time interval.
How to Check the Mind
The acceleration may change the speed module, direction, or both properties at the same time. Then check the units, vector direction and behaviour of the result at zero or very large parameter.
Σ
Conclusion on the topic
Vector kinematics
The position of the material point sets the radius-vector, the change is displacement. The instantaneous velocity is directed tangential to the trajectory, the acceleration describes the change of the velocity vector.
The average and instantaneous speeds differ in the limit transition to a short time interval.
The acceleration may change the speed module, direction or both at the same time.
Any numerical result shall be accompanied by a unit, a sign and an explanation of its physical meaning.
The content is consistent with the FRP on Physics of the OS/OSS 2025 and the subject matter of the current lines of the PNP: I. M. Peryškin and co-authors (7–9), G. J. Mäkishev and coauthors (10–11), and with the lines "Engineers of the Future" and V. A. Kasyanov. The original texts are not reproduced, the text of the textbooks is not reproduced.