Kinematics is a branch of mechanics that focuses on describing the motion of objects without considering the forces that cause it. It examines parameters like position, velocity, and acceleration to analyze and predict the trajectories of particles and rigid bodies. By using concepts such as displacement, speed, and relative motion, this section provides tools to understand and solve problems involving motion paths and rates, forming a foundation for studying more complex dynamics in engineering and physics.
Considering a vector \mathbf r with a magnitude and a direction, the derivatives with respect of time is defined as:
\frac{\mathrm d \mathbf r}{\mathrm dt} \equiv \dot {\mathbf r}
so either or both can change as time passes.
When taking vector derivatives, it is necessary to specific a reference frame because the time derivative of a vector might differ in different reference frames: for example, considering a fixed vector on a rotating disc, when considering a position staying on the disc, the time derivatives is equal to zero, while considering a reference frame outside the disc, the same vector has a time derivatives that is not equal to zero.
A reference frame is a coordinate system or viewpoint used to measure and observe the position, orientation, and motion of objects. It provides a basis for describing physical quantities such as displacement, velocity, and acceleration, typically relative to a fixed or moving point in space.
Considering a generic vector \mathbf r = r_1 \mathbf e_1 + r_2 \mathbf e_2 + r_3 \mathbf e_3, and a reference frame RF, the time derivative of the vector with respect to that frame is:
\frac{\mathrm d \mathbf r}{\mathrm dt}\bigg|_{RF} = \sum_{i=1}^3 \left(\frac{\mathrm d r_i}{\mathrm dt}\mathbf e_i + r_i \frac{\mathrm d \mathbf e_i}{\mathrm dt}\bigg|_{RF} \right)
Considering a point P, the position is defined as the vector which goes from the origin of a reference frame to P:
\mathbf r_P
The velocity is the derivative of the position with respect of time:
\mathbf v_P \equiv \frac{\mathrm d \mathbf r_P}{\mathrm dt} = \dot {\mathbf r}_P
The acceleration is the derivative of the velocity with respect of time:
\mathbf a_P \equiv \frac{\mathrm d \mathbf v_P}{\mathrm dt} =\dot {\mathbf v}_P = \frac{\mathrm d^2 \mathbf r_P}{\mathrm dt^2} = \ddot {\mathbf r}_P
To derive the velocity from the acceleration, it is necessary to integrate it:
\mathbf v_P = \int \mathbf a_P \mathrm dt
And integrate again to get the position:
\mathbf r_{OP} = \int \mathbf v_P \mathrm dt = \iint \mathbf a_P \mathrm dt^2
Rectilinear motion refers to the movement of an object along a straight line. In this type of motion, all parts of the object travel in parallel paths with constant or varying speed and acceleration.
The key variables describing rectilinear motion include displacement, velocity, and acceleration, which can be functions of time, but, since they are on a straight line, they are all scalar quantities:
\begin{aligned} v(t) & = v_0 + \int_0^t a(\tau) \, d\tau \\ r(t) & = r_0 + \int_0^t v(\tau) \, d\tau \end{aligned}
There are several different coordinate systems, the most commonly used for curvilinear is the Cartesian for which the orthonormal vectors are \mathbf i, \mathbf j and \mathbf k. Since these are fixed, the position, velocity and acceleration are:
\begin{aligned} \mathbf r_P & = x \mathbf i + y \mathbf j + z \mathbf k \\ \mathbf v_P & = \dot x \mathbf i + \dot y \mathbf j + \dot z \mathbf k \\ \mathbf a_P & = \ddot x \mathbf i + \ddot y \mathbf j + \ddot z \mathbf k \\ \end{aligned}
Another commonly used system for curvilinear motion is the cylindrical coordinate system described by three unitary vectors \mathbf e_\rho, \mathbf e_\phi, \mathbf e_z. The position, velocity, and acceleration of a point P are described by the coordinates (\rho, \phi, z), where \rho is the radial distance, \phi is the azimuthal angle, and z is the height along the z-axis. The unit vectors in this system are \mathbf{e}_\rho, \mathbf{e}_\phi, and \mathbf{e}_z, which vary with position due to the curvilinear nature of the system.
The expressions for position, velocity, and acceleration in cylindrical coordinates are:
\begin{aligned} \mathbf{r}_P &= \rho \mathbf{e}_\rho + z \mathbf{e}_z \\ \mathbf{v}_P &= \dot{\rho} \mathbf{e}_\rho + \rho \dot{\phi} \mathbf{e}_\phi + \dot{z} \mathbf{e}_z \\ \mathbf{a}_P &= \left( \ddot{\rho} - \rho \dot{\phi}^2 \right) \mathbf{e}_\rho + \left( 2 \dot{\rho} \dot{\phi} + \rho \ddot{\phi} \right) \mathbf{e}_\phi + \ddot{z} \mathbf{e}_z \\ \end{aligned}
In the spherical coordinates system, the position, velocity, and acceleration are described by the coordinates (r, \theta, \phi), where r is the radial distance, \theta is the polar angle, and \phi is the azimuthal angle. The corresponding unit vectors are \mathbf{e}_r, \mathbf{e}_\theta, and \mathbf{e}_\phi, which also vary with position.
The expressions for position, velocity, and acceleration in spherical coordinates are:
\begin{aligned} \mathbf{r}_P &= r \mathbf{e}_r \\ \mathbf{v}_P &= \dot{r} \mathbf{e}_r + r \dot{\theta} \mathbf{e}_\theta + r \sin \theta \, \dot{\phi} \mathbf{e}_\phi \\ \mathbf{a}_P &= \left( \ddot{r} - r \left( \dot{\theta}^2 + \sin^2 \theta \, \dot{\phi}^2 \right) \right) \mathbf{e}_r + \left( 2 \dot{r} \dot{\theta} + r \ddot{\theta} - r \sin \theta \cos \theta \, \dot{\phi}^2 \right) \mathbf{e}_\theta + \left( 2 \dot{r} \sin \theta \, \dot{\phi} + 2 r \dot{\theta} \cos \theta \, \dot{\phi} + r \sin \theta \, \ddot{\phi} \right) \mathbf{e}_\phi \\ \end{aligned}
These formulations provide a complete description of motion in Cartesian, cylindrical, and spherical coordinate systems, with each system suited for specific types of curvilinear motion.
The Cartesian, cylindrical, and spherical coordinate systems described above are extrinsic coordinate systems, meaning they are defined with respect to fixed axes in space. In contrast, intrinsic coordinate systems, such as the tangential and normal coordinate systems, are defined based on the motion of the particle itself. This makes intrinsic systems particularly useful for describing curvilinear motion along a path or trajectory.
In the tangential and normal (or Frenet-Serret) coordinate system, we define the motion of a point P along a path using the following unit vectors:
The position vector is generally described by the path or trajectory itself, while the velocity and acceleration can be expressed in terms of the intrinsic coordinates:
\begin{aligned} \mathbf{v}_P &= v \, \mathbf{e}_t \\ \mathbf{a}_P &= \frac{d v}{d t} \, \mathbf{e}_t + \frac{v^2}{\rho} \, \mathbf{e}_n \\ \end{aligned}
where:
The binormal component of the acceleration is zero if the motion is confined to a plane. However, in three-dimensional motion, the binormal component of the acceleration vector would be governed by the torsion of the path and the derivative of \mathbf{e}_b.