Course Title: Dynamics of a Particle
Course Code: DYN 233
Rectilinear Kinematics: Continuous
Motion
Objectives
1. To introduce the concepts of position, displacement, velocity, and acceleration.
2. To study the continuous rectilinear motion of a particle and represent this motion graphically.
3. To distinguish between average and instantaneous kinematic quantities.
1 Introduction
Mechanics is that branch of the physical sciences concerned with the behavior of bodies subjected
to the action of forces. The subject of mechanics is divided into two parts:
Statics - the study of objects in equilibrium (objects either at rest or moving with a constant
velocity).
• Dynamics - the study of objects in motion.
• The subject of dynamics is often itself divided into two parts:
o kinematics - treats only the geometric aspects of the motion.
o kinetics - analysis of forces causing the motion.
• In this chapter, we study the kinematics of a particle - recall that a particle has a mass but
negligible size and shape. Therefore, we limit discussion to those objects that have dimensions
that are of no consequence in the analysis of the motion. Such objects may be considered as
particles, provided motion of the body is characterized by motion of its mass center and any
rotation of the body is neglected.
1.1 Rectilinear Kinematics: Continuous Motion
• Rectilinear Kinematics refers to straight line motion. The kinematics of a particle is
characterized by specifying, at any given instant, the particle’s position, velocity, and
acceleration.
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allowed. For more of these, contact: . stuvia
, o Position. The position of the particle is represented by an algebraic scalar’s (the
position coordinate).
o Displacement. The displacement of the particle is a vector ∆𝐫 defined as the change in
the particle’s position vector 𝐫.
o Velocity. The velocity of the particle is a vector.
∆𝐫
• The average velocity is the displacement divided by time i.e., 𝐯𝑎𝑣𝑔 = ∆𝑡.
𝑑𝐫
• The instantaneous velocity is 𝐯 = 𝑑𝑡.
𝑑𝑠
• Speed refers to the magnitude of velocity and is written as 𝑣 = |𝐯| = 𝑑𝑡.
• Average speed is the total distance divided by the total time (different from average velocity
which is displacement divided by time).
𝑑𝐯
o Acceleration. The acceleration of the particle is a vector 𝐚 = 𝑑𝑡 . Its magnitude is
𝑑𝑣 𝑑2 𝑠
written as 𝑎 = = .
𝑑𝑡 𝑑𝑡 2
• In rectilinear kinematics, the acceleration is negative when the particle is slowing down or
decelerating.
• A particle can have an acceleration and yet have zero velocity.
𝑑𝑣 𝑑𝑠
• The relationship 𝑎𝑑𝑠 = 𝑣𝑑𝑣 is derived from 𝑎 = 𝑑𝑡
and 𝑣 = 𝑑𝑡 by eliminating 𝑑𝑡.
CONSTANT ACCELERATION
• Let 𝑎 = 𝑎𝑐 = constant. Assume that 𝑣 = 𝑣0 and 𝑠 = 𝑠0 at time 𝑡 = 0. Then
𝑣 = 𝑣0 + 𝑎𝑐 𝑡 (speed as a function of time) 1. 1
1
𝑠 = 𝑠0 + 𝑣0 𝑡 + 2 𝑎𝑐 𝑡 2 , (position as a function of time) 1. 2
𝑣 2 = 𝑣0 2 + 2𝑎𝑐 (𝑠 − 𝑠0 ) (speed as a function of position) 1. 3
PROCEDURE FOR SOLVING PROBLEMS
The equations of rectilinear kinematics should be applied as follows:
• Coordinate System
o Establish a position coordinate s along the path and specify its fixed origin and positive
direction.
o Since motion is along a straight line, the particle’s position, velocity, and acceleration can
be represented as algebraic scalars. For analytical work, the sense of 𝑠, 𝑣 and 𝑎 is then
determined from their algebraic signs.
o The positive sense for each scalar can be indicated by an arrow shown alongside each
kinematic equation as it is applied.
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
, • Kinematic Equations.
o If a relationship is known between any two of the four variables 𝑎, 𝑣, 𝑠 and 𝑡, then a third
𝑑𝑣 𝑑𝑠
variable can be obtained by using one of the kinematic equations 𝑎 = 𝑑𝑡
, 𝑣 = 𝑑𝑡 or
𝑎𝑑𝑠 = 𝑣𝑑𝑣, which relates all three variables.
o Whenever integration is performed, it is important that the position and velocity be
known at a given instant to evaluate either the constant of integration if an indefinite
integral is used, or the limits of integration if a definite integral is used.
o Note that Eqs. (1.1)–(1.3) apply only when the acceleration is constant.
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
, Summary Notes
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
Course Code: DYN 233
Rectilinear Kinematics: Continuous
Motion
Objectives
1. To introduce the concepts of position, displacement, velocity, and acceleration.
2. To study the continuous rectilinear motion of a particle and represent this motion graphically.
3. To distinguish between average and instantaneous kinematic quantities.
1 Introduction
Mechanics is that branch of the physical sciences concerned with the behavior of bodies subjected
to the action of forces. The subject of mechanics is divided into two parts:
Statics - the study of objects in equilibrium (objects either at rest or moving with a constant
velocity).
• Dynamics - the study of objects in motion.
• The subject of dynamics is often itself divided into two parts:
o kinematics - treats only the geometric aspects of the motion.
o kinetics - analysis of forces causing the motion.
• In this chapter, we study the kinematics of a particle - recall that a particle has a mass but
negligible size and shape. Therefore, we limit discussion to those objects that have dimensions
that are of no consequence in the analysis of the motion. Such objects may be considered as
particles, provided motion of the body is characterized by motion of its mass center and any
rotation of the body is neglected.
1.1 Rectilinear Kinematics: Continuous Motion
• Rectilinear Kinematics refers to straight line motion. The kinematics of a particle is
characterized by specifying, at any given instant, the particle’s position, velocity, and
acceleration.
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
, o Position. The position of the particle is represented by an algebraic scalar’s (the
position coordinate).
o Displacement. The displacement of the particle is a vector ∆𝐫 defined as the change in
the particle’s position vector 𝐫.
o Velocity. The velocity of the particle is a vector.
∆𝐫
• The average velocity is the displacement divided by time i.e., 𝐯𝑎𝑣𝑔 = ∆𝑡.
𝑑𝐫
• The instantaneous velocity is 𝐯 = 𝑑𝑡.
𝑑𝑠
• Speed refers to the magnitude of velocity and is written as 𝑣 = |𝐯| = 𝑑𝑡.
• Average speed is the total distance divided by the total time (different from average velocity
which is displacement divided by time).
𝑑𝐯
o Acceleration. The acceleration of the particle is a vector 𝐚 = 𝑑𝑡 . Its magnitude is
𝑑𝑣 𝑑2 𝑠
written as 𝑎 = = .
𝑑𝑡 𝑑𝑡 2
• In rectilinear kinematics, the acceleration is negative when the particle is slowing down or
decelerating.
• A particle can have an acceleration and yet have zero velocity.
𝑑𝑣 𝑑𝑠
• The relationship 𝑎𝑑𝑠 = 𝑣𝑑𝑣 is derived from 𝑎 = 𝑑𝑡
and 𝑣 = 𝑑𝑡 by eliminating 𝑑𝑡.
CONSTANT ACCELERATION
• Let 𝑎 = 𝑎𝑐 = constant. Assume that 𝑣 = 𝑣0 and 𝑠 = 𝑠0 at time 𝑡 = 0. Then
𝑣 = 𝑣0 + 𝑎𝑐 𝑡 (speed as a function of time) 1. 1
1
𝑠 = 𝑠0 + 𝑣0 𝑡 + 2 𝑎𝑐 𝑡 2 , (position as a function of time) 1. 2
𝑣 2 = 𝑣0 2 + 2𝑎𝑐 (𝑠 − 𝑠0 ) (speed as a function of position) 1. 3
PROCEDURE FOR SOLVING PROBLEMS
The equations of rectilinear kinematics should be applied as follows:
• Coordinate System
o Establish a position coordinate s along the path and specify its fixed origin and positive
direction.
o Since motion is along a straight line, the particle’s position, velocity, and acceleration can
be represented as algebraic scalars. For analytical work, the sense of 𝑠, 𝑣 and 𝑎 is then
determined from their algebraic signs.
o The positive sense for each scalar can be indicated by an arrow shown alongside each
kinematic equation as it is applied.
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
, • Kinematic Equations.
o If a relationship is known between any two of the four variables 𝑎, 𝑣, 𝑠 and 𝑡, then a third
𝑑𝑣 𝑑𝑠
variable can be obtained by using one of the kinematic equations 𝑎 = 𝑑𝑡
, 𝑣 = 𝑑𝑡 or
𝑎𝑑𝑠 = 𝑣𝑑𝑣, which relates all three variables.
o Whenever integration is performed, it is important that the position and velocity be
known at a given instant to evaluate either the constant of integration if an indefinite
integral is used, or the limits of integration if a definite integral is used.
o Note that Eqs. (1.1)–(1.3) apply only when the acceleration is constant.
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
, Summary Notes
© 2026 Prepared by Core Axis. Redistribution, reuploading, resale, or public posting without permission is not
allowed. For more of these, contact: . stuvia
