Stephen M. Birn and Bela I. Sandor

There are two major categories in dynamics, kinematics and kinetics.
Kinematics involves the time- and geometry-dependent motion of a particle,
rigid body, deformable body, or a ?uid without considering the forces that
cause the motion. It relates position, velocity, acceleration, and time. Kinetics
combines the concepts of kinematics and the forces that cause the motion.

Kinematics of Particles
Scalar Method

The scalar method of particle kinematics is adequate for one-dimensional
analysis. A particle is a body whose dimensions can be neglected
(in some analyses, very large bodies are considered particles).
The equations described here are easily adapted and applied to two and
three dimensions.

Average and Instantaneous Velocity

The  average  velocity of a particle is the change in distance divided by
the change in time.  The instantaneous velocity is the particles velocity
at a particular instant.

Average and Instantaneous Acceleration

The  average acceleration is the change in  velocity divided by
the change in time. The instantaneous acceleration is the particles
acceleration at a particular instant.

Displacement, velocity, acceleration, and time are related to one another.
For example, if velocity is given as a function of time, the displacement and
acceleration can be determined through integration and differentiation,
respectively. The following example illustrates this concept.

Example 8

A particle moves with a velocity v(t) = 3t
2 8t. Determine x(t) and a(t), if x(0) = 5.
Solution.
1. Determine x(t) by integration

2. Determine a(t) by differentiation

There are four key points to be seen from these graphs (Figure 1.3.1).

FIGURE 1.3.1 Plots of a particles kinematics.

1. v = 0 at the local maximum or minimum of the x-t curve.
2. a = 0 at the local maximum or minimum of the v-t curve.
3. The area under the v-t curve in a speci?c time interval is equal to the
net displacement change in that interval.
4. The area under the a-t curve in a speci?c time interval is equal to the
net velocity change in that interval.

Useful Expressions Based on Acceleration

Equations for nonconstant acceleration:

Equations for constant acceleration (projectile motion; free fall):

These equations are only to be used when the acceleration is known to
be a constant. There are other expressions available depending on how
a variable acceleration is given as a function of time, velocity,
or displacement.

Scalar Relative Motion Equations

The concept of relative motion can be used to determine the displacement,
velocity, and acceleration between two particles that travel along the same line.
Equation 1.3.6 provides the mathematical basis for this method. These
equations can also be used when analyzing two points on the same body
that are not attached rigidly to each other (Figure 1.3.2).

FIGURE 1.3.2 Relative motion of two particles along a straight line.

The notation B/A  represents the displacement, velocity, or acceleration
of particle B as seen from particle A. Relative motion can be used to
analyze many different degrees-of-freedom systems. A degree of freedom
of a mechanical system is the number of independent coordinate systems
needed to de?ne the position of a particle.

Vector Method

The vector method facilitates the analysis of two- and three-dimensional
problems. In general, curvilinear motion occurs and is analyzed using
a convenient coordinate system.

Vector Notation in Rectangular (Cartesian) Coordinates

Figure 1.3.3 illustrates the vector method.

FIGURE 1.3.3 Vector method for a particle.

The mathematical method is based on determining v and a as functions of the
position vector r. Note that the time derivatives of unit vectors are zero
when the xyz coordinate system is ?xed. The scalar components 
(x?, y, ?x,….)  can be determined from the appropriate scalar equations
previously presented that only include the quantities relevant to the coordinate
direction considered.

There are a few key points to remember when considering curvilinear motion.
First, the instantaneous velocity  vector is  always tangent to the path
of the particle. Second, the speed of the particle is the magnitude of the velocity
vector. Third, the acceleration vector is not tangent to the path of the particle
and not collinear with v in curvilinear motion.

Tangential and Normal Components

Tangential and normal components are useful in analyzing  velocity
and acceleration.  Figure 1.3.4 illustrates the method and Equation 1.3.8
is the governing equations for it.

? = r =   constant for a circular path

FIGURE 1.3.4 Tangential and normal components.
C is the center of curvature.

The osculating plane contains the unit vectors nt and nn, thus de?ning a plane.
When using normal and tangential components, it is common to forget to
include the component of normal acceleration, especially if the particle travels
at a constant speed along a curved path. For a particle that moves in
circular motion,

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