01 - Kinematics

1. Position and Trajectory

Reference System & Point Particle

Motion is defined relative to a reference system: a set of bodies whose relative positions do not vary, with respect to which motion is measured.

Key Definition

When describing the motion of an object, you must always specify the reference system first.

A point particle is an abstraction that retains the physical properties (mass, charge, etc.) of the original object but ignores its size. A point particle can only translate; extended bodies can also rotate, making their description more complex.

---

Trajectory and Hourly Law (Motion Law)

• Trajectory $s=s(x,y)$: the set of positions occupied by the point during motion — geometrically, a curve in space.
• Motion law (hourly law) $s=s(t)$: describes how the position varies with time.

Trajectory vs. Motion Law

The trajectory is purely geometric (the path shape). The motion law adds the time dimension, telling you when the point is at each position.

By setting an origin $O$, we measure the arc length $s$ traveled along the trajectory as a function of time.

---

Dimensions of Motion

Type	Coordinates needed	Example
1D (unidimensional)	1	Motion along a straight line
2D (bidimensional)	2	Motion on a plane
3D (three-dimensional)	3	Motion in space

The trajectory is a vector of corresponding size.

---

Medicine & Biology Applications

Clinical Relevance Position measurements and trajectories are increasingly important in medicine:

• CT, MRI, ultrasound, PET: localize structures with mm-level spatial resolution (voxels/pixels).
• ECG: electrical activity of the heart as $f(t)$ → detects arrhythmias, fibrillations.
• Functional MRI: monitors brain activity changes over time.
• Serial CT in radiotherapy: tracks tumor size and shape over treatment course.

We can generalize “trajectory” to mean the variation of any parameter $x$ over time:

$x=x(t)$

Examples: temperature, heart rate, protein expression level, cell proliferation rate — before and after perturbations (pathology, surgery, therapy).

---

2. Velocity and Acceleration

Mean Scalar Velocity

${\bar{v}}_s=\frac{\Delta s}{\Delta t}=\frac{s(t_2)-s(t_1)}{t_2-t_1}$

Dimensions: $[v]=\frac{L}{T}$, SI unit: $m/s$.

---

Mean Vector Velocity

$\overrightarrow{\stackrel{ˉ}{v}}=\frac{\Delta \vec{s}}{\Delta t}=\frac{\vec{s}(t_2)-\vec{s}(t_1)}{t_2-t_1}$

The modulus of the mean

vector velocity equals the mean scalar velocity only in 1D motion. In multi-dimensional motion they may differ.

---

Instantaneous Velocity

$v={\lim }_{\Delta t\to 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt}{|}_{t=t_1}$

The instantaneous velocity vector $\vec{v}$ is always tangent to the trajectory at the current point:

$\vec{v}=\frac{ds}{dt},{\hat{u}}_{tg}$

The velocity vector can vary in modulus, direction, and sense over time: $\vec{v}=\vec{v}(t)$.

• Instantaneous velocity → tangent to the trajectory.
• Average velocity → secant between points $s_1$ and $s_2$.

---

Acceleration

When velocity varies over time, the motion is accelerated.

${\vec{a}}_m=\frac{\Delta \vec{v}}{\Delta t}=\frac{{\vec{v}}_1-{\vec{v}}_0}{t_1-t_0}$

$\vec{a}={\lim }_{\Delta t\to 0}\frac{\Delta \vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}$

Acceleration in Common Language vs. Physics

In physics, acceleration occurs whenever the velocity vector changes — in magnitude or direction:

• A car speeding up → accelerating ✓
• A car braking → accelerating ✓ (deceleration is negative acceleration)
• A car turning at constant speed → accelerating ✓ (direction changes)

Brakes and steering wheels are both “accelerators” in the physical sense.

For straight-line motion, acceleration is along the trajectory:

$a=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}$

For curved motion, $\vec{a}$ lies in the plane of the trajectory and points toward the concavity.

---

3. Fundamental 1D Motions

Multi-dimensional motions are combinations of these fundamental 1D motions (e.g., parabolic motion = uniform horizontal + uniformly accelerated vertical).

---

Uniform Rectilinear Motion

Constant speed along a straight line; $a(t)=0$.

$\boxed{s(t)=s_0+v_0,t}$

Biomedical Examples Uniform rectilinear motion arises when the net force is zero (friction balances propulsion):

• Free fall in air (terminal velocity reached)
• Particle sedimentation or centrifugation in a viscous medium
• Charged particles in a conductor (current) or in electrophoresis
• Wave propagation in a uniform medium (constant speed, though direction may change)

---

Uniformly Accelerated Rectilinear Motion

Constant acceleration $a(t)=a_0$.

$\boxed{v(t)=v_0+a_0,t}$

$\boxed{s(t)=s_0+v_0,t+\frac{1}{2}{a}_0,t^2}$

From eliminating $t$, velocity as a function of displacement:

$v^2(t)=v_0^2+2a_0(s(t)-s_0)$

Characteristics

• (a) Constant acceleration
• (b) Velocity varies linearly with time
• (c) Position varies quadratically with time

---

Harmonic Motion

The simplest oscillatory motion, arising when the restoring force is proportional to displacement from equilibrium.

$\boxed{s(t)=A\sin (\omega t+\phi )}$

Symbol	Name	Meaning
$A$	Amplitude	Maximum displacement from equilibrium
$\omega$	Angular frequency (pulsation)	Oscillations in $2\pi$ seconds
$\phi$	Initial phase	Starting angle at $t=0$
$\nu$	Frequency	Complete oscillations per second; $\omega =2\pi \nu$
$T$	Period	Time for one complete oscillation; $T=1/\nu$

Initial position: $s_0=A\sin (\phi )$, so $\sin (\phi )=\frac{s_0}{A}$ (fraction of amplitude at $t=0$).

Velocity and acceleration:

$v(t)=\frac{ds}{dt}=A\omega \cos (\omega t+\phi )$

$a(t)=\frac{d^{2}s}{d{t}^2}=-A{\omega }^2\sin (\omega t+\phi )=-{\omega }^{2}s$

Acceleration is proportional to displacement and directed opposite to it. Harmonic motion is the first-order approximation of any perturbation near a stable equilibrium (minimum of potential energy).

---

4. Multi-Dimensional Kinematics

Position Vector

In 3D, position is described by the vector $\vec{r}$ with components $(r_x,r_y,r_z)$ projected onto the $x$, $y$, $z$ axes.

Everything from 1D extends naturally: $\vec{r}$, $\vec{v}$, $\vec{a}$ are all 3-component vectors.

---

Tangential and Normal Acceleration

In curved motion, acceleration splits into two components:

$a_t=|\vec{a}|\cos \theta =\frac{d^{2}s}{d{t}^2}\text{(tangential — changes speed)}$

$a_n=|\vec{a}|\sin \theta =\frac{1}{R}{\left(\frac{ds}{dt}\right)}^2\text{(normal — changes direction)}$

where $R$ is the radius of curvature of the trajectory at that point.

Three Causes of Acceleration in 2D/3D

a. Change in modulus only (same direction)
b. Change in direction only (same modulus)
c. Change in both modulus and direction

---

Uniform Circular Motion

Point moves along a circumference with constant speed modulus; acceleration is constant in modulus and always directed toward the center (centripetal).

With $\theta (t)$ = angle swept, $R$ = radius, angular velocity $\omega =d\theta /dt$:

$s(t)=R,\theta (t)$

$v=\frac{ds}{dt}=\omega R$

$|\vec{a}|={\omega }^{2}R\text{(centripetal)}$

---

Ballistic (Projectile) Motion — Galileo

Decomposed into two independent motions:

$\vec{a}=-g,{\hat{u}}_y⟹a_x=0,a_y=-g$

Velocity components:

$v_x(t)=v_{x0},v_y(t)=v_{y0}-g,t$

Position:

$x(t)=x_0+v_{x0},t,y(t)=y_0+v_{y0},t-\frac{1}{2}g,t^2$

Trajectory (eliminating $t$):

$y(x)=y_0+\frac{v_{y0}}{v_{x0}}(x-x_0)-\frac{g}{2v_{x0}^2}(x-x_0{)}^2$

This is a parabola.

At the apex (maximum height, $v_y=0$):

$t_a=\frac{v_{y0}}{g},x_a=x_0+v_{x0}\cdot \frac{v_{y0}}{g},y_a=y_0+\frac{v_{y0}^2}{2g}$

Speed modulus at any point: $v=\sqrt{v_x^2+v_y^2}$; direction is tangent to the parabola.

If given initial speed $v_0$ and elevation angle ${\theta }_0$:

$v_{x0}=v_0\cos {\theta }_0,v_{y0}=v_0\sin {\theta }_0$

$\tan \alpha =\frac{v_y}{v_x}\text{(instantaneous angle of velocity)}$

---

TL;DR

Complete Summary — Kinematics

Core concepts

• Motion is always defined relative to a reference system. A point particle is a simplification retaining mass/charge but ignoring size (translation only; no rotation).
• The trajectory $s(x,y)$ is the geometric path; the motion law $s(t)$ adds time information.
• Motion can be 1D, 2D, or 3D (1, 2, or 3 coordinates).

Velocity

• Mean scalar: $\bar{v}=\Delta s/\Delta t$
• Instantaneous: $v=ds/dt$ — always tangent to the trajectory.
• Vector velocity can vary in magnitude, direction, and sense.

Acceleration

• $a=dv/dt=d^{2}s/d{t}^2$; a vector directed toward the concavity of the trajectory.
• In physics, any change in the velocity vector (braking, turning, speeding up) constitutes acceleration.

The 3 fundamental 1D motions (building blocks for all others)

1. Uniform rectilinear: $a=0$; $s(t)=s_0+v_{0}t$ — straight line, constant speed. Examples: sedimentation, electrophoresis at terminal velocity, wave propagation.
2. Uniformly accelerated rectilinear: $a=\text{const}$; $v(t)=v_0+at$; $s(t)=s_0+v_{0}t+\frac{1}{2}a{t}^2$; $v^2=v_0^2+2a\Delta s$.
3. Harmonic: $s(t)=A\sin (\omega t+\phi )$; $a=-{\omega }^{2}s$ (restoring, proportional to displacement). First approximation of any oscillation near stable equilibrium.

Multi-dimensional kinematics

• Position/velocity/acceleration become 3-component vectors.
• Acceleration splits into tangential $a_t=d^{2}s/d{t}^2$ (changes speed) and normal $a_n=v^2/R$ (changes direction).
• Uniform circular motion: constant speed on a circle; centripetal acceleration $a={\omega }^{2}R$ directed toward center.
• Ballistic motion: superposition of uniform horizontal motion + uniformly accelerated vertical motion → parabolic trajectory. Apex at $t_a=v_{y0}/g$.

Medical relevance: kinematics of any measurable parameter over time (ECG, fMRI signal, tumor size in serial CT, protein expression) can be analyzed with the same mathematical tools — trajectory = parameter vs. time curve.