In this chapter, we apply the time derivative presented in Chapter 1 to study the kinematics of the point (mass point or particle), which is the simplest model for material bodies.

The mass point model is not suitable when the orientation is relevant to the problem under study. The simplest model incorporating the orientation is the rigid body, which is defined as a set of material points with constant mutual distances. This definition is so close to that of the reference frame that both concepts are taken as synonyms in a kinematics context. The reference frame of points fixed with respect to the rigid body is called body reference frame. The kinematics of the rigid body corresponds to the transportation motion (Chapter 2) associated with that reference frame.

Kinematics deals with the geometry of motion of material bodies along time – change of position in space and over time – without regard to the physical phenomena on which it depends. Such description requires mathematical models for space and time (which is the physical framework of mechanical phenomena) and mathematical models for bodies.

The study of the dynamics of a mechanical system starts with the description of its mechanical state (position and velocity of every point in the system). For the particular case of systems consisting of a finite number of rigid bodies, though the number of points is infinite, that description calls for a finite set of position variables – generalized coordinates (GC) – and speed variables – generalized speeds (GS). The vector space defined by the GC is called configuration space; that defined by the GC and the GS is known as phase space.

Scientists have long studied fire in an effort to both understand the world around them and to prevent the destruction and devastation that uncontrolled fires can cause. Despite many advances in the understanding of fire phenomena, society offers continued challenges that require new approaches for the prevention and mitigation of unwanted fires. In this chapter, fire research is presented through a series of photographs that scale from small, buoyant flames in the laboratory up to large, uncontrolled wildfires and even fire whirls.

There are only so many technologies and devices that have the same type of impact as that of the internal combustion (IC) engine. Its ubiquitous nature pervades our everyday life, many times without us even realizing it. Whether it be the spark-ignited engine driving our vehicle, the compression-ignition engine hauling food to our local grocery store, the jet engine we hear flying 38,000 feet overhead, or the gas turbine powering the laptop screen from which we read this article, internal combustion engines are quite literally intricately and irreplaceably woven into our daily lives. The internal combustion has taken on many different forms throughout its long, greater than 150-year history, but combustion has always been one of its few constants. Indeed, combustion is even in its name and helps differentiate it from other thermodynamic work devices such heat engines and fuel cells.

Combustion is a critically important and extremely visual phenomenon. When properly controlled, combustion is important for a wide range of processes. For example, it is the primary source of power generation for our vast array of electrical equipment and electronics. However, when combustion is not properly controlled, it can be a source of great devastation. For example, uncontrolled wildfires are still a major concern in many parts of the world.

Our perception of a flame is strongly grounded in gravity’s influence. From our every interaction with fire from the first birthday candles we blew out, we each build an intuitive understanding of how a flame interacts with the hot air rising via buoyant convection. As researchers, our perceptions of how flames respond to our controls are unconsciously biased by this intrinsic buoyant flow.