Dynamics: Kinematics of Particles

Kinematics is the branch of dynamics that describes motion without asking what causes it. For a particle (an object whose size can be neglected relative to the motion of interest), kinematics focuses on four core ideas: position, displacement, velocity, and acceleration. The same physical motion can be described in different coordinate systems, and choosing the right one is often the difference between a clean solution and a messy one.

This article develops particle kinematics for rectilinear and curvilinear motion, shows how to express motion in Cartesian, polar, and path (tangential-normal) coordinates, and closes with relative motion, which is essential whenever motion is observed from a moving frame.

What kinematics describes

A particle’s position is specified by a position vector $\mathbf{r}(t)$ measured from a chosen origin. Motion is defined by how $\mathbf{r}$ changes with time:

• Displacement: $\Delta \mathbf{r}=\mathbf{r}(t_2)-\mathbf{r}(t_1)$
• Velocity: $\mathbf{v}=\frac{d\mathbf{r}}{dt}$
• Acceleration: $\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{d{t}^2}$

Two distinctions matter early on:

• Distance traveled vs displacement: distance depends on path length; displacement depends only on endpoints.
• Speed vs velocity: speed is the magnitude $v=\Vert \mathbf{v}\Vert$; velocity is a vector with direction.

Rectilinear motion (straight-line motion)

Rectilinear motion occurs when a particle moves along a straight line. In one dimension, take the coordinate $x(t)$ along that line:

• Velocity: $v=\frac{dx}{dt}$
• Acceleration: $a=\frac{dv}{dt}=\frac{d^{2}x}{d{t}^2}$

Constant acceleration formulas

When acceleration is constant (common in introductory problems and useful for approximations), the standard relations follow from integration:

• $v(t)=v_0+at$
• $x(t)=x_0+v_{0}t+\frac{1}{2}a{t}^2$
• $v^2=v_0^2+2a(x-x_0)$

These equations work only when acceleration is constant and motion remains along one line. Even then, sign convention matters. If “positive” is chosen upward, gravitational acceleration near Earth becomes $a=-g$.

Acceleration as a function of velocity or position

Not all rectilinear motion uses constant acceleration. Sometimes acceleration is given as a function of velocity, $a(v)$, or position, $a(x)$. Two useful identities help:

• If $a=\frac{dv}{dt}$ and $v=\frac{dx}{dt}$, then

$$a=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}$$

This form is especially useful when time is not explicitly present, such as braking problems where acceleration depends on position along a track.

Curvilinear motion (motion along a curve)

Most real motion is curvilinear: vehicles turning, projectiles in flight, satellites in orbit. The particle’s position changes in both magnitude and direction, and the acceleration generally has components that reflect both changes.

There are three widely used coordinate descriptions:

• Cartesian $(x,y,z)$
• Polar (or cylindrical in 3D) $(r,\theta )$
• Path coordinates (tangential-normal) $(t,n)$

Each is valid; the best choice is dictated by the geometry of the motion.

Cartesian coordinates: component-based description

In Cartesian coordinates, write $$\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$$ Then differentiate component-wise: $$\mathbf{v}=\dot{x}\mathbf{i}+\dot{y}\mathbf{j}+\dot{z}\mathbf{k}$$ $$\mathbf{a}=\ddot{x}\mathbf{i}+\ddot{y}\mathbf{j}+\ddot{z}\mathbf{k}$$

Cartesian form is ideal when motion is naturally described by independent component equations, such as projectile motion with known $x(t)$ and $y(t)$, or when acceleration components are specified directly (for example, constant $a_x$ and $a_y$).

Polar coordinates: radial and transverse motion

For planar motion that naturally references a point (like circular motion about a center), polar coordinates are often clearer. Position is described by radius $r(t)$ and angle $\theta (t)$.

The velocity and acceleration have standard forms in the radial and transverse directions:

• Velocity:

$$\mathbf{v}=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }$$

• Acceleration:

$$\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2){\mathbf{e}}_r+(r\ddot{\theta }+2\dot{r}\dot{\theta }){\mathbf{e}}_{\theta }$$

Interpretation matters:

• $(\ddot{r}-r{\dot{\theta }}^2)$ is the radial acceleration component. The term $-r{\dot{\theta }}^2$ appears even if $r$ is constant; it represents the inward (centripetal) part associated with changing direction.
• $(r\ddot{\theta }+2\dot{r}\dot{\theta })$ is the transverse component. The $2\dot{r}\dot{\theta }$ term shows up when the radius is changing while the particle also rotates, a coupling that is easy to miss in Cartesian work.

Special case: circular motion

For uniform circular motion, $r=R$ constant and $\dot{\theta }=\omega$ constant:

• Speed: $v=R\omega$
• Acceleration magnitude: $a=R{\omega }^2=\frac{v^2}{R}$ directed toward the center.

Even though the speed is constant, acceleration is not zero because the velocity direction is continuously changing.

Path coordinates: tangential and normal components

When the path geometry is known or curvature is central to the problem, it is often best to describe motion using the tangent and normal directions along the trajectory.

Let $s(t)$ be the arc length measured along the path. Then:

• Speed: $v=\frac{ds}{dt}$
• Tangential acceleration:

$$a_t=\frac{dv}{dt}$$

• Normal (centripetal) acceleration:

$$a_n=\frac{v^2}{\rho }$$

Here $\rho$ is the radius of curvature of the path at that point. This decomposition separates two physically distinct effects:

• $a_t$ changes the magnitude of velocity (speeding up or slowing down).
• $a_n$ changes the direction of velocity (turning), even if speed is constant.

A practical example is vehicle motion on a curved road: drivers feel the normal acceleration as lateral “cornering” demand, while tangential acceleration corresponds to throttle or braking along the lane.

Relative motion: describing motion from moving observers

Kinematics becomes especially powerful when comparing motion between observers. Relative motion answers questions like: How fast is one vehicle approaching another? What is the velocity of a boat relative to the shore given a current?

Relative position, velocity, and acceleration

Let particle $A$ and particle $B$ have position vectors ${\mathbf{r}}_A$ and ${\mathbf{r}}_B$ measured from the same inertial origin. Define the position of $A$ relative to $B$ as: $${\mathbf{r}}_{A/B}={\mathbf{r}}_A-{\mathbf{r}}_B$$ Differentiate to obtain: $${\mathbf{v}}_{A/B}={\mathbf{v}}_A-{\mathbf{v}}_B$$ $${\mathbf{a}}_{A/B}={\mathbf{a}}_A-{\mathbf{a}}_B$$

This framework is simple but widely applicable. If two cars move in the same direction, subtracting velocities gives the closing speed. If two aircraft cross paths, relative velocity provides the rate and direction at which separation changes.

Translating frames: a common engineering assumption

When the reference frame attached to $B$ is translating but not rotating relative to an inertial frame, the subtraction relationships above hold directly in vector form. This is often the assumption in introductory relative motion problems, such as “plane relative to wind” or “swimmer relative to river current,” where velocities combine by vector addition: $${\mathbf{v}}_{\text{object/ground}}={\mathbf{v}}_{\text{object/medium}}+{\mathbf{v}}_{\text{medium/ground}}$$

Rotation of the reference frame introduces additional terms (beyond basic particle kinematics), so it is important to know whether the observer is merely moving or also turning.

Choosing the right coordinate system

A reliable workflow for particle kinematics is:

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