Chapter 1: Kinematics

Kinematics is a branch of physics founded in classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without taking into account the forces that cause them to move. A kinematics problem begins with the description of the system's geometry and the declaration of initial conditions for any known values of position, velocity, and/or acceleration of points inside the system. Using geometry arguments, the position, velocity, and acceleration of any unknown system components can then be computed. Kinematics does not investigate how forces operate on bodies; kinetics does. For further information, consult analytical dynamics.

In astrophysics, kinematics is used to describe the motion of celestial entities and groups of such bodies. Kinematics is utilized in mechanical engineering, robotics, and biomechanics to describe the motion of multi-link systems, such as an engine, a robotic arm, or the human skeleton.

Geometric transformations, also known as rigid transformations, are utilized to characterize the motion of mechanical system components, hence facilitating the derivation of the equations of motion. Moreover, they are fundamental to dynamic analysis.

Kinematic analysis is the measurement of kinematic quantities used to describe motion. In engineering, for example, kinematic analysis can be used to determine the range of motion for a given mechanism, whereas kinematic synthesis can be used to create a mechanism with the desired range of motion. Moreover, kinematics uses algebraic geometry to examine the mechanical advantage of a mechanical system or mechanism.

Kinematic is the English translation of A.M.

Ampère's cinématique, The field of particle kinematics studies the trajectory of particles. A particle's position is defined as the vector of coordinates from the origin of a coordinate frame to the particle. Consider a tower 50 m south of your home. If the coordinate frame is centered at your home, with east along the x-axis and north along the y-axis, the coordinate vector to the tower's base is r = (0 m, 50 m, 0 m). If the tower is 50 m tall and its height is measured along the z-axis, then the vector of coordinates to the top of the tower is r = (0 m, 50 m, 50 m).

A three-dimensional coordinate system is used to determine the position of a particle in the most general scenario. If the particle is confined to a plane, however, a two-dimensional coordinate system is adequate. All physics observations are insufficient if not stated relative to a reference frame.

A particle's location vector is a vector drawn from the reference frame's origin to the particle.

It indicates both the point's distance from the origin and its direction away from the origin.

Three dimensionally, the position vector

can be expressed as

where

,

, and

are the Cartesian coordinates and

,

and

are the unit vectors along the

,

, and

coordinate axes, respectively.

The magnitude of the position vector

gives the distance between the point

and the origin.

The direction is quantified by the direction cosines of the position vector. In general, the position vector of an item is dependent on the frame of reference; different frames will result in different position vector values.

Particle trajectory is a vector function of time,

, which specifies the trajectory tracked by a moving particle, given by

where

,

, and

describe each coordinate of the particle's position as a function of time.

The velocity of a particle is a vector quantity that specifies both the direction and magnitude of the particle's motion. Mathematically, the velocity of a point is the rate of change of its position vector with respect to time. Consider the ratio created by dividing the difference between two particle positions by the time interval. This proportion is referred to as the average velocity throughout that span of time and is defined as

where

is the change in the position vector during the time interval

.

In the limit that the time interval

approaches zero, The average speed approaches the instantaneous speed, defined as the position vector's time derivative,

where the dot signifies a time-dependent derivative (e.g.

).

Thus, The velocity of a particle is the rate of change in its position over time.

Furthermore, This velocity is tangent to the trajectory of the particle at every point along its route.

A reference frame that is not spinning, Because the directions and magnitudes of the coordinate directions are constant, their derivatives are disregarded.

The magnitude of an object's velocity is its speed. It is a scalar value:

where

is the arc-length measured along the trajectory of the particle.

This arc-length must increase continuously as the particle moves.

Hence,

is non-negative, This implies that velocity is also positive.

The velocity vector may vary in magnitude, direction, or both simultaneously. Consequently, acceleration accounts for both the rate of change in magnitude and direction of the velocity vector. The same logic used to establish velocity based on the position of a particle may also be used to calculate acceleration based on velocity. The acceleration of a particle is the vector defined by the velocity vector's rate of change. The average acceleration of a particle over a time interval is defined as the ratio of the particle's initial acceleration to its final acceleration.

where ?v is the difference in the velocity vector and ?t is the time interval.

As the time interval approaches 0, the acceleration of the particle approaches the limit of the average acceleration, which is the time derivative,

or

Therefore, acceleration is the first derivative of the particle's velocity vector and the second derivative of its position vector. As their directions and magnitudes are constant in a non-rotating frame of reference, the derivatives of the coordinate directions are not taken into account.

The magnitude of an object's acceleration is the magnitude of its acceleration vector, denoted by |a|. It is a scalar value:

A relative position vector is a vector that defines the relative position of two points. It is the positional difference between the two locations. The location of point A with respect to another point B just represents the difference between their places.

where is the difference between their position vector components.

If point A has position components

and point B has position components

then the position of point A relative to point B is the difference between their components:

The relative velocity of two points is just the difference in their velocities.

This is the difference between their velocity components.

If point A has velocity components

and point B has velocity components

then the velocity of point A relative to point B is the difference between their components:

Alternatively, this same result could be obtained by computing the time derivative of the relative position vector rB/A.

In cases where the velocity is close to the speed of light c (usually within 95 percent), special relativity employs another system of relative velocity called rapidity, which is dependent on the ratio of v to c.

The acceleration of point C relative to point B is just the difference between their respective accelerations.

This is the difference between their acceleration components.

If point C has acceleration components

and point B has acceleration components

then the acceleration of point C relative to point B is the difference between their components:

Alternatively, this same result could be obtained by computing the second time derivative of the relative position vector rB/A.

Assuming that the position's basic criteria are met,

, and velocity

at time

are known, The initial integration produces the particle's velocity as a function of time.

A second integration reveals its course (trajectory),

It is possible to derive further relationships between displacement, velocity, acceleration, and time. Since acceleration is constant,

can be inserted into the equation above to yield:

Without explicit time dependency, a relationship can be established between velocity, position, and acceleration by solving the average acceleration for time, substituting, and simplifying.

where

denotes the dot product, which is proper because the outcomes are scalars and not vectors.

The dot product can be replaced by the cosine of the angle a between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in such a case:

In the case of acceleration, it must always be in the direction of motion, whether the motion is positive or negative, the angle between the vectors (a) is 0, so

, and

This can be simplified using the notation for the magnitudes of the vectors

where

can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so

This reduces the particle's parametric equations of motion to a Cartesian relationship of velocity against position.

This relationship is beneficial when...