Course Title: Dynamics of a Particle
Course Code: DYN 233
Plane Curvilinear Motion: Rectangular
Components
Objectives
1. Define and interpret the position, velocity, and acceleration vectors for plane curvilinear motion
using time derivatives.
2. Express and analyze motion in rectangular components, including computation of scalar components
and directions.
3. Distinguish between the magnitude of a derivative and the derivative of a magnitude in vector
motion.
4. Apply rectangular coordinate methods to solve projectile motion problems under constant
gravitational acceleration.
1 Curvilinear Motion
Plane curvilinear motion describes the motion of a particle along a curved path confined to a single
plane. Unlike rectilinear motion, the particle’s position, velocity, and acceleration vectors may change in
both magnitude and direction. The summary introduces the time derivative of a vector as a fundamental
concept in dynamics and develops rectangular component analysis for solving practical engineering
problems, including projectile motion.
1.1 Superposition of rectilinear motions
Curvilinear motion can be treated as a superposition of independent rectilinear motions when the
acceleration components are uncoupled, meaning each component depends only on its own coordinate,
velocity component, and time.
𝑎𝑥 = 𝑓𝑥 (𝑣𝑥 , 𝑥, 𝑡)
𝑎𝑦 = 𝑓𝑦 (𝑣𝑦 , 𝑦, 𝑡) 1.1
𝑎𝑧 = 𝑓𝑧 (𝑣𝑧 , 𝑧, 𝑡)
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
, The motions in the x-, y-, and z-directions can then be analyzed separately using rectilinear motion
principles. Motion in a plane for instance is the superposition of two independent rectilinear motions, such
as projectile motion under constant gravity.
• Uncoupled Motion. Acceleration in each coordinate direction depends only on its
corresponding variables, allowing independent analysis of x-, y-, and z-motions.
• Coupled Motion. When acceleration components depend on multiple coordinates or
velocity components, the equations become coupled and generally require numerical
methods for solution.
1.2 General Curvilinear Motion
If Eqs.(1.1) are coupled, an analytical solution will be difficult or impossible. Equations of this type
must be solved by numerical methods which are outside of the scope of this summary note.
2 Plane Curvilinear Motion
The vast majority of the motions of points or particles encountered in engineering practice can be
represented as plane motion. Plane curvilinear motion describes the motion of a particle along a curved
path confined to a single plane. The motion can be treated using vector analysis, independent of any
particular coordinate system, since the results will be independent of any particular coordinate system. What
follows in this article constitutes one of the most basic concepts in dynamics, namely, the time derivative
of a vector.
2.1 Position
Consider now the continuous motion of a particle along a plane curve as represented in
The position of the particle is described by the position vector 𝐫. This vector is a function of time
since both its magnitude and direction change as the particle moves along its path (described by the path
function 𝑠).
Displacement over time 𝛥𝑡 is Δ𝐫, while the scalar distance traveled is 𝛥𝑠.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
, The rectangular component of position is
𝐫 = 𝑥𝐢 + 𝑦𝐣
2.2 Velocity
The average velocity between two positions is defined as
𝐯ave = Δ𝐫/Δ𝑡
The instantaneous velocity is the limiting value as Δt → 0,
𝐯 = lim Δ𝐫/Δ𝑡
𝛥𝑡→0
Thus, the velocity of the particle is described by the vector
𝑑𝐫
𝐯= = 𝐫̇
𝑑𝑡
To get the rectangular components of the velocity,
𝑑𝐫 𝑑[𝑥𝐢 + 𝑦𝐣]
𝐯= =
𝑑𝑡 𝑑𝑡
𝑑 𝑑
= [𝑥𝐢] + [𝑦𝐣]
𝑑𝑡 𝑑𝑡
𝑑 𝑑
= [𝑥]𝐢 + [𝑦]𝐣
𝑑𝑡 𝑑𝑡
= 𝑣𝑥 𝐢 + 𝑣𝑦 𝐣
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
Course Code: DYN 233
Plane Curvilinear Motion: Rectangular
Components
Objectives
1. Define and interpret the position, velocity, and acceleration vectors for plane curvilinear motion
using time derivatives.
2. Express and analyze motion in rectangular components, including computation of scalar components
and directions.
3. Distinguish between the magnitude of a derivative and the derivative of a magnitude in vector
motion.
4. Apply rectangular coordinate methods to solve projectile motion problems under constant
gravitational acceleration.
1 Curvilinear Motion
Plane curvilinear motion describes the motion of a particle along a curved path confined to a single
plane. Unlike rectilinear motion, the particle’s position, velocity, and acceleration vectors may change in
both magnitude and direction. The summary introduces the time derivative of a vector as a fundamental
concept in dynamics and develops rectangular component analysis for solving practical engineering
problems, including projectile motion.
1.1 Superposition of rectilinear motions
Curvilinear motion can be treated as a superposition of independent rectilinear motions when the
acceleration components are uncoupled, meaning each component depends only on its own coordinate,
velocity component, and time.
𝑎𝑥 = 𝑓𝑥 (𝑣𝑥 , 𝑥, 𝑡)
𝑎𝑦 = 𝑓𝑦 (𝑣𝑦 , 𝑦, 𝑡) 1.1
𝑎𝑧 = 𝑓𝑧 (𝑣𝑧 , 𝑧, 𝑡)
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
, The motions in the x-, y-, and z-directions can then be analyzed separately using rectilinear motion
principles. Motion in a plane for instance is the superposition of two independent rectilinear motions, such
as projectile motion under constant gravity.
• Uncoupled Motion. Acceleration in each coordinate direction depends only on its
corresponding variables, allowing independent analysis of x-, y-, and z-motions.
• Coupled Motion. When acceleration components depend on multiple coordinates or
velocity components, the equations become coupled and generally require numerical
methods for solution.
1.2 General Curvilinear Motion
If Eqs.(1.1) are coupled, an analytical solution will be difficult or impossible. Equations of this type
must be solved by numerical methods which are outside of the scope of this summary note.
2 Plane Curvilinear Motion
The vast majority of the motions of points or particles encountered in engineering practice can be
represented as plane motion. Plane curvilinear motion describes the motion of a particle along a curved
path confined to a single plane. The motion can be treated using vector analysis, independent of any
particular coordinate system, since the results will be independent of any particular coordinate system. What
follows in this article constitutes one of the most basic concepts in dynamics, namely, the time derivative
of a vector.
2.1 Position
Consider now the continuous motion of a particle along a plane curve as represented in
The position of the particle is described by the position vector 𝐫. This vector is a function of time
since both its magnitude and direction change as the particle moves along its path (described by the path
function 𝑠).
Displacement over time 𝛥𝑡 is Δ𝐫, while the scalar distance traveled is 𝛥𝑠.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
, The rectangular component of position is
𝐫 = 𝑥𝐢 + 𝑦𝐣
2.2 Velocity
The average velocity between two positions is defined as
𝐯ave = Δ𝐫/Δ𝑡
The instantaneous velocity is the limiting value as Δt → 0,
𝐯 = lim Δ𝐫/Δ𝑡
𝛥𝑡→0
Thus, the velocity of the particle is described by the vector
𝑑𝐫
𝐯= = 𝐫̇
𝑑𝑡
To get the rectangular components of the velocity,
𝑑𝐫 𝑑[𝑥𝐢 + 𝑦𝐣]
𝐯= =
𝑑𝑡 𝑑𝑡
𝑑 𝑑
= [𝑥𝐢] + [𝑦𝐣]
𝑑𝑡 𝑑𝑡
𝑑 𝑑
= [𝑥]𝐢 + [𝑦]𝐣
𝑑𝑡 𝑑𝑡
= 𝑣𝑥 𝐢 + 𝑣𝑦 𝐣
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: and go to YT: @engora-ph stuvia
