ME 101: Engineering Mechanics
Semester 2, 2025/2026
Chapter 12 - Kinematics of a Particle - Part 1
Lecturer: Bernard Osei Adomako | Faculty Intern: Keziah Enyonam Noamesi
30/03/2026
Objectives
1. To introduce the concepts of position, displacement, velocity,and acceleration.
2.To study particle motion along a straight line and represent this motion graphically.
3.To investigate particle motion along a curved path using different coordinate systems.
4.To present an analysis of dependent motion of two particles.
5.To examine the principles of relative motion of two particles using translating axes.
Introduction to Mechanics
Mechanics is a branch of the physical sciences that is concerned with the
state of rest or motion of bodies that are subjected to the action of forces.
Mechanics
Rigid Body
(Rigid-Body Mechanics)
Statics
Kinematics
Dynamics
(Deals with motion without
reference to the cause of motion)
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Deformable Body
(Solid Mechanics)
Strength of
Materials
Kinetics
Theory of
Elasticity
(Deals with motion with reference
to the cause of motion)
Fluid
(Fluid Mechanics)
Incompressible
Compressible
Theory of
Plasticity
Introduction to Mechanics
Dynamics
Kinematics
Deals with only the geometric
aspects of the motion
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(Deals with accelerated motion of bodies,
thus time dependent)
Kinetics
Deals with the analysis of the
forces causing the motion.
Problem Solving
Dynamics is considered to be more involved than statics since both the forces applied to a
body and its motion must be taken into account. Also, many applications require using
calculus, rather than just algebra and trigonometry. In any case, the most effective way of
learning the principles of dynamics is to solve problems. To be successful at this, it is
necessary to present the work in a logical and orderly manner as suggested by the following
sequence of steps:
Read the problem carefully and try to correlate the actual physical situation with the
theory studied.
Tabulate the problem data and draw to a large scale any necessary diagrams.
Apply the relevant principles, generally in mathematical form. When writing any equations,
be sure they are dimensionally homogeneous.
Solve the necessary equations, and report the answer with no more than three significant
figures.
Study the answer with technical judgment and common sense to determine whether or
not it seems reasonable.
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Rectilinear Kinematics: the kinematics of a particle that moves along a straight path. The kinematics of a
particle is characterized by specifying, at any given instant, the particle’s position, velocity, and
acceleration.
Position: A particle travel along a straight-line path defined by the coordinate axis ‘s’. The position of the
particle relative to the origin, O, is defined by the position vector s, or the scalar s. Scalar s can be
positive or negative. s and s are usually measured in metres (m) or feet (ft).
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Displacement: The displacement of the particle is defined as the change in its position. For example, if
the particle moves from one point to another, the displacement is
.Displacement is also a
vector quantity, and it should be distinguished from the distance the particle travels. Specifically, the
distance traveled is a positive scalar that represents the total length of path over which the particle
travels.
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Velocity: Velocity is a measure of the rate of change in the position of a particle. It is a vector quantity
(it has both magnitude and direction). The magnitude of the velocity is called speed, with units of m/s or
ft/s. If the particle moves through a displacement Δs during the time interval Δt, the average velocity of
the particle is
Instantaneous Velocity: This refers to velocity at a specific instant. It is obtained as the limit of average
velocity as Δt → 0. It is given by:
Direction is determined by displacement since time, Δt is always positive:
Positive → motion to the right
Negative → motion to the left
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
The average speed is always a positive scalar and is defined as the total distance traveled by a particle,
,
divided by the elapsed time Δt, ie.,
The particle in the figure travels along the path of
length sT in time Δt, so its average speed is
(Vavg)sp = sT/Δt, but its average velocity is
Vavg =- Δs/Δt
Acceleration: Acceleration is the time rate of change of velocity. It is a vector quantity.
Average Acceleration: the rate at which velocity changes over a given time interval.
Δv represents the difference in the velocity during the time interval Δt, i.e., Δv = v’ - v,
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Instantaneous Acceleration: This refers to acceleration at a specific instant or moment in time (not
over an interval). It is obtained by taking smaller and smaller values of Δt and corresponding smaller
and smaller values of Δv. It is given by:
Also,
Both the average and instantaneous acceleration can be positive or negative.
Positive acceleration → velocity increasing in same direction
Negative acceleration (deceleration) → velocity decreasing or acting opposite motion
Relationship between displacement, velocity, and acceleration:
(obtained from the chain rule)
Units for the magnitude of acceleration are:
Note: A particle can have zero velocity but nonzero acceleration because acceleration measures
the rate of change of velocity, not the value of velocity at that instance.
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RECTILINEAR KINEMATICS: CONTINUOUS MOTION
At constant acceleration, we can obtain formulas that relate the constant acceleration to v, s and t.
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Types of Kinematics Problems
Acceleration specified as a function of time
Acceleration specified as a constant
Acceleration specified as a function of velocity
Acceleration specified as a function of position
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Example
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Solution
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Example
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Solution
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Example
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Solution
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Example
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Example
RECTILINEAR KINEMATICS: ERRATIC MOTION
When a particle has erratic or changing motion, then its position, velocity, and acceleration cannot
be described by a single continuous mathematical function along the entire path. This type of
motion is described using using graphs and piecewise relationships.
There are five types of graphs that can be used to describe this type of motion.
Position vs Time Graph
Velocity vs Time Graph
Acceleration vs Time Graph
Velocity vs Position Graph
Acceleration vs Position Graph
Building Graphs (Concept Flow)
From s–t → v–t → take slope
From v–t → a–t → take slope
From a–t → v–t → take area
From v–t → s–t → take area
Slope = derivative
Area = integral
RECTILINEAR KINEMATICS: ERRATIC MOTION
Position vs Time Graph, s–t Graph
Slope of s-t graph = velocity
Velocity vs Time Graph, v–t Graph
Slope of v-t graph = acceleration
Acceleration vs Time Graph, a–t Graph
Area under a-t graph = change in velocity
To construct the v-t graph, we begin with the particle’s initial
velocity v0 and then v1 = v0 + Δv, v2 = v0 + Δv, vi = v(i -1) + Δv
Area above axis → increase in velocity
Area below axis → decrease in velocity
RECTILINEAR KINEMATICS: ERRATIC MOTION
Velocity vs Time Graph, v–t Graph
Area under v-t graph = displacement
Acceleration vs Position Graph, a–s Graph
Area under a-s graph:
Connects acceleration
velocity squared
with
change
in
Velocity vs Position Graph, v–s Graph
Acceleration can be found from slope:
Acceleration = velocity × slope of v-s graph
RECTILINEAR KINEMATICS: ERRATIC MOTION
Example
RECTILINEAR KINEMATICS: ERRATIC MOTION
Solution
RECTILINEAR KINEMATICS: ERRATIC MOTION
Example
RECTILINEAR KINEMATICS: ERRATIC MOTION
Solution
RECTILINEAR KINEMATICS: ERRATIC MOTION
Example
RECTILINEAR KINEMATICS: ERRATIC MOTION
Solution
GENERAL CURVILINEAR MOTION
Curvilinear motion occurs when a particle moves along a curved path. Since this path is often
described in three dimensions, vector analysis will be used to formulate the particle’s position,
velocity, and acceleration.
General aspects of curvilinear motion
Key quantities:
Position → r(t)
Velocity → v
Acceleration → a
1.Position & Displacement
Position Vector: The position of a particle on a space
curve defined by the path function s(t), measured from a
fixed point O, will be designated by the position vector r =
r(t).
Displacement: Displacement Δr represents the change in
the particle’s position and is determined by vector
subtraction; i.e., Δr = r’ - r.
GENERAL CURVILINEAR MOTION
2.Velocity
Average velocity:
Instantaneous velocity:
Important:
Velocity is always tangent to the path
Speed (magnitude):
Note: speed can be obtained by differentiating the path function ‘s’
with respect to time.
3.Acceleration
GENERAL CURVILINEAR MOTION
Average acceleration:
Instantaneous acceleration:
Important:
Velocity is always tangent to the path
Also:
Hodograph & Key Insight
Hodograph: path traced by the velocity vector tip
Acceleration: Tangent to the hodograph, however
not necessarily tangent to the motion path.
CURVILINEAR MOTION
Curvilinear motion occurs when a particle moves along a curved path. Since this path is often
described in three dimensions, vector analysis will be used to formulate the particle’s position,
velocity, and acceleration.
Three Coordinate Systems to Consider for Curvilinear Motion:
1.Cartesian or rectangular components, (x, y, z)
2.Natural (normal + tangent)
3.Polar, 2D (r, θ)/ Cylindrical, 3D (r, θ, z)
CURVILINEAR MOTION: Rectangular Components
Position: If the particle is at point (x, y, z) on the path shown in the figure below, then its location is
defined by the position vector
,where i, j, k are unit vectors
At any instant the magnitude and direction of r can be calculated as:
;
Meaning:
r tells how far the particle is from the origin
ur gives the direction from the origin to the particle
Velocity: Velocity is the time rate of change of position. Hence,
CURVILINEAR MOTION: Rectangular Components
Differentiation of Vector Components (Velocity in Rectangular Coordinates):
The magnitude and direction of velocity
is given by:
;
CURVILINEAR MOTION: Rectangular Components
Acceleration: Acceleration is the time rate of change of velocity.
where;
The magnitude and direction of acceleration is given by:
;
Note: a in general will not be tangent to the path
CURVILINEAR MOTION: Rectangular Components
CURVILINEAR MOTION: Rectangular Components
Example
CURVILINEAR MOTION: Rectangular Components
Solution
CURVILINEAR MOTION: Rectangular Components
Example
CURVILINEAR MOTION: Rectangular Components
Solution
MOTION OF A PROJECTILE
A projectile is any object that is thrown, launched, kicked, or fired into the air and then moves
under the effect of gravity only.
A projectile’s motion is studied by splitting it into two separate motions:
1. Horizontal motion
2. Vertical motion
These two happen at the same time, but they behave differently.
Horizontal Motion (horizontal acceleration is zero)
Vertical Motion (a constant acceleration of ‘-g’ is assumed
MOTION OF A PROJECTILE
Example
MOTION OF A PROJECTILE
Solution
MOTION OF A PROJECTILE
Example
MOTION OF A PROJECTILE
Solution
MOTION OF A PROJECTILE
Example
MOTION OF A PROJECTILE
Solution
MOTION OF A PROJECTILE