5.1: Space, time and motion
We start by introducing the concepts of space, time and motion, the building blocks of kinematics.
In space the position of each point mass \(i\) can be described by a position vector \(\overrightarrow{\boldsymbol{r}}_{i}\). Space can be parametrised by a three-dimensional (3D) coordinate system as introduced in Sec. 3.3. This space, in which the shortest distance between two points is a straight line, is also called Euclidean space.
The position of objects in space can change \({ }^{1}\) in time \(t\).
The motion \(\overrightarrow{\boldsymbol{r}}_{i}(t)\) of a single point-like object \(i\) is a function that describes its position coordinates at every time \(t\) (see Figure 5.1). The motion of a system of \(N\) point masses is represented by a set of vector functions \(\overrightarrow{\boldsymbol{r}}_{1}(t), \overrightarrow{\boldsymbol{r}}_{2}(t), \ldots\) \(, \overrightarrow{\boldsymbol{r}}_{i}(t), \ldots, \overrightarrow{\boldsymbol{r}}_{N}(t)\) that describe the positions of all points \(i\) in the system at all times.
Kinematics describes the motion of objects in time and space, irrespective of the forces that cause them to move. Kinematic techniques allow one to determine the relations between position, velocity and acceleration vectors, and to apply constraint equations for motion analysis.
Kinematic information on a point mass is provided in two ways:
- Full or partial information on the time dependence of the position, velocity or acceleration vectors or its components.
- Constraint equations that pose certain limitations on how the point mass can move.
The goal of kinematics is to analyse the given information to obtain more detailed information on the motion of the particle. The main challenges and skills in kinematics we will address in this chapter are:
- To describe the motion of a point mass using a time-dependent position vector \(\overrightarrow{\boldsymbol{r}}(t)\) in a suitable coordinate system.
- To relate and determine position, velocity and acceleration vectors from each other.
- To use constraint equations and equations of motion to determine and predict the time-dependent motion \(\overrightarrow{\boldsymbol{r}}(t)\) of point masses.