MECHANICS the initial position 𝑠𝑜, the displacement is
negative.
LECTURE 6: RECTILINEAR TRANSLATION ● The displacement is a vector quantity. This
means it has a direction as well as a
❖ DYNAMICS magnitude and is represented visually as an
arrow that points from the initial position to the
Dynamics- is the branch of mechanics which deals final position.
with the study of bodies in motion. The experiments
which form the foundation of dynamics require the ❖ VELOCITY
use of three kinds of units: force, length, and time
It is divided into two branches called kinetics and Velocity- is defined as a vector measurement of the
kinematics. rate and direction of motion; the speed at which
something moves in one direction. The speed of a car
1. Kinematics- is the geometry of motion and traveling north on a major freeway and the speed a
used to define the motion of a particle or body rocket launching into space can both be measured
without consideration of the forces causing the using velocity.
motion. (e.g. displacement, velocity, and
acceleration) ● In calculus terms, velocity is the first derivative
2. Kinetics- is the analysis of the forces causing of position with respect to time. You can
the motion. It relates the force acting on a calculate velocity by using a simple formula
body to its mass and acceleration. that includes rate, distance, and time.
The motion of the particle may either be rectilinear
motion or curvilinear motion.
● Rectilinear motion- refers to the particle as it
moves along a straight line while a particle in ● The average velocity of the particle over the
a Curvilinear motion- moves along a curved time interval ∆𝑡 is defined as the quotient of
line in two or three dimensions. the displacement ∆𝑠 and the time interval ∆𝑡
as
Application of Dynamics in Engineering
1. Structural design of any vehicle such as
automobile or airplane.
2. Design of mechanical devices such as motors,
pumps, movable tools, industrial manipulators, The average velocity is expressed in meters per
and machinery second (m/s) or in feet per second (ft/s).
3. Predictions of the motions of artificial
satellites, projectiles, and spacecraft. ● The instantaneous velocity v of a particle at
the instant t is determined by allowing the time
Important Terms in Dynamics interval Δt to become infinite similarly small.
● Body - It denotes a system of particles which Thus,
form an object of appreciable size.
● Particle - It usually denotes an object of point
size. (Singer). It has a mass but negligible
size and shape.
The instantaneous velocity is also expressed in m/s
● Position- It is used to specify the location of a
or ft/s.
particle at any given instant.
● The most common way to calculate the
❖ DISPLACEMENT
constant velocity of an object moving in a
straight line is with this formula:
Displacement- is defined to be the change in position
of an object. Say, a particle moves from the initial
position 𝑠𝑜 to a final position 𝑠𝑓, then the displacement
is the difference in distance between the two positions
NOTE:
The SI (international) units for velocity are m/s
● If the final position 𝑠𝑓 is to the right of the (meters per second), but velocity may also be
initial position 𝑠𝑜, the displacement is positive. expressed in any units of distance per time [e.g. miles
Likewise, if the final position 𝑠𝑓 is to the left of per hour (mph), kilometers per hour (kph), and
kilometers per second (km/s)].
❖ Difference Between Speed and Velocity Deceleration- is used to refer to a when the speed of
the particle (i.e., the magnitude of v) decreases; the
Speed- is a scalar quantity that indicates the rate of particle is then moving more slowly.
motion distance per time. Its units are length and Another way to define acceleration is,
time. Speed is often described simply as the distance
traveled per unit of time. It is how fast an object is
moving.
Velocity- is a vector quantity that gives the rate of
motion of a particle in a certain direction.
❖ Relationship Between Velocity &
Acceleration
Velocity and Acceleration are all related to each other,
though they represent different measurements.
Velocity is a vector quantity that indicates
displacement, time, and direction while acceleration
is the rate at which an object changes its velocity.
❖ RECTILINEAR MOTION OF PARTICLES
Rectilinear motion- is another name for
❖ Acceleration straight-line motion. This type of motion describes
the movement of a particle or a body.
Acceleration- is defined as a vector quantity that - A body is said to experience rectilinear motion
indicates the rate of change of velocity. It has if any two particles of the body travel the same
dimensions of length and time over time. distance along two parallel straight lines.
- is often referred to as “speeding up”, but it Following are the rectilinear motion examples:
really measures changes in velocity. An object ● Use of elevators in public places is an
is accelerating if it is changing its velocity. example of rectilinear motion.
- can be experienced every day in a vehicle. ● Gravitational forces acting on objects resulting
You step on the accelerator and the car in free fall is an example of rectilinear motion.
speeds up, increasing its velocity. ● Kids sliding down from a slide is a rectilinear
motion.
● If the velocity of the particle is known at any ● Motion of planes in the sky is a rectilinear
two points, the average acceleration during motion.
the time interval ∆t is defined as:
There are three types of rectilinear motion and they
are:
1. Uniform rectilinear motion
The average acceleration is expressed in m/s2 or in 2. Uniformly accelerated rectilinear motion
ft/s2 3. Rectilinear movement with non-uniform
acceleration
● The instantaneous acceleration a of the The three-basic equation of rectilinear motion are as
particle at the instant t is obtained by again follows
allowing the time interval ∆𝑡 to approach zero.
● The instantaneous acceleration at time t is a
vector that is found by taking the smaller and
smaller values of ∆𝑡 and corresponding small
values of ∆𝑣 , so that:
The instantaneous acceleration is also expressed in
𝑚/𝑠^2 or 𝑓𝑡/𝑠^2 .
LECTURE 7: MOTION OF A PROJECTILE Horizontal Range and Maximum Height
- The Horizontal Range is maximum at an initial
❖ Projectile angle of 45 °
Projectile- Is any object thrown into space upon
which the only acting force is the gravity. In other
words, the primary force acting on a projectile is
gravity. This doesn’t necessarily mean that the other
forces do not act on it, just that their effect is minimal
compared to gravity.
● Trajectory- The path followed by a projectile.
Ex. A baseball batted or thrown and the instant the
bullet exits the barrel of a gun. It is divided into two
branches called kinetics and kinematics.
● A projectile is any object that once projected
or dropped continues in motion by its own
inertia and is influenced only by the downward
force of gravity.
LECTURE 8: FORCE AND ACCELERATION
When a particle is thrown obliquely near the earth’s
surface, it moves along a curved path under constant
Newton’s Second Law of Motion: Force and
acceleration that is directed towards the center of the
Acceleration
earth (we assume that the particle remains close to
● A particle of mass m acted upon by an
the surface of the earth). The path of such a particle is
unbalanced force F experiences an
called a projectile and the motion is called projectile
acceleration a that has the same direction as
motion. Air resistance to the motion of the body is to
the force and a magnitude that is directly
be assumed absent in projectile motion.
proportional to the force.
● The acceleration of an object is directly
In a projectile Motion, there are two simultaneous
proportional to the net force on it and inversely
independent rectilinear motions:
proportional to its mass.
1. Along x-axis: uniform velocity, responsible for
● It means Greater mass = Greater amount of
the horizontal (forward) motion of the particle.
force needed to accelerate
2. Along y-axis: uniform acceleration,
responsible for the vertical (downwards)
motion of the particle.
It states that the rate of change of momentum of a
body is directly proportional to the impressed force
and it takes place in the direction of the force acting
on it.
● Therefore, can analyze projectile motions as ● 1 Newton is the amount of force that will
combination of horizontal motion with constant accelerate a 1kg mass at the rate of 1 m/s2
velocity and vertical motion with constant ● So 1N = 1 kg • m/s2
acceleration. Weight is the force due to gravity.
● Using these concepts and setting 𝑌𝑜=0 and
𝑋0=0, we have the equations
