CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.1 INTRODUCTION AND OVERVIEW OF COURSE CONTENT
Objectives:
• Define various terms relative to the course (kinematics, kinetics, particle).
• Understand the usefulness of particle dynamics.
• Find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle
traveling along a straight path.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.2 RECTILINEAR KINEMATICS
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1: Establish Kinematic Equations for Constant Acceleration
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
Initially, the car travels along a straight road with a speed of 35 m/s. If the brakes are applied and
the speed of the car is reduced to 10 m/s in 15 s, determine the constant deceleration of the car.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 3
A particle travels along a straight line such that its acceleration is = (4𝑡 ! − 2) 𝑚/𝑠 ! , where 𝑡 is
in seconds. When 𝑡 = 0, the particle is located 2 𝑚 to the left of the origin, and when 𝑡 = 2 𝑠,
it is 20 𝑚 to the left of the origin. Determine the position of the particle when 𝑡 = 4 𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. In dynamics, a particle is assumed to have _________.
a) both translation and rotational motions
b) only a mass
c) a mass but the size and shape cannot be neglected
d) no mass or size or shape, it is just a point
2. The average speed is defined as ________.
∆#
a) ∆$
#
b) $
c) 𝑠% /∆𝑡
d) None of the above
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.3 ERRATIC MOTION IN RECTILINEAR KINEMATICS
Objectives:
• Determine position, velocity, and acceleration of a particle using graphs.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1
If a particle starts from rests and accelerates according to the graph shown, find the particle’s
velocity at 𝑡 = 20𝑠 and the time at which the particle stops moving.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
The car shown starts from rest and travels along a straight track
such that it accelerates at 10𝑚/𝑠 ! for 10s, and then decelerates at
2𝑚/𝑠 ! . Draw graphs demonstrating the velocity vs. time and
position vs. time. Determine the time needed to stop the car and
how far the car travels in total.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. The slope of a v-t graph at any instant represents instantaneous:
a) Velocity
b) Acceleration
c) Position
d) Jerk
2. Displacement of a particle over a given time interval equals the area under the ____ graph
during that time.
a) a-t
b) a-s
c) v-t
d) s-t
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.4 CURVILINEAR MOTION
Objectives:
• Describe the motion of a particle traveling along a curved path.
• Related kinematic quantities in terms of the rectangular components of the vectors.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.5 CURVILINEAR MOTION – RECTANGULAR COMPONENTS
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1
A particle P moves along the x-axis. Derive its’ position, velocity and acceleration vectors.
Ex. 2
A particle travels along a straight line path 𝑦 = 0.5𝑥. If the x component of the particles velocity
is 𝑣& = 2𝑡 ! 𝑚/𝑠, determine the magnitude of the particles velocity and acceleration when 𝑡 = 4𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 3
A particle travels along the path 𝑦 = 0.5𝑥 ! and 𝑥 = 𝑦 = 𝑧 = 0 when 𝑡 = 0. If the horizontal
component of the particles velocity is 𝑣& = 5𝑡 𝑓𝑡/𝑠, determine the particles distance and the
magnitude of its acceleration when 𝑡 = 1𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 4
A box slides down a slope described by the equation 𝑦 = 0.05𝑥 ! 𝑚, 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑖𝑠 𝑖𝑛 𝑚𝑒𝑡𝑒𝑟𝑠.
Additionally, at 𝑥 = 5𝑚 the horizontal components of the velocity and acceleration are 𝑣& =
−3𝑚/𝑠, 𝑎& = −1.5𝑚/𝑠 ! . Determine the corresponding vertical components of the velocity and
acceleration at 𝑥 = 5𝑚.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. In curvilinear motion, the direction of the instantaneous velocity is always:
a) Perpendicular to the path
b) Tangent to the path
c) Perpendicular to the instantaneous acceleration
d) Tangent to the instantaneous acceleration
2. If the position of a particle is defined by 𝒓 = (1.5𝑡 ! + 1)𝚤̂ + (4𝑡 − 1)𝚥̂ 𝑚, what is its speed
at 𝑡 = 1 𝑠?
a) 2m/s
b) 3m/s
c) 5m/s
d) 7m/s
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.6 PROJECTILE MOTION
Objectives:
• Analyze the free-flight motion of a projectile.
Consider a projectile 𝑷 launched at (𝒙𝟎 , 𝒚𝟎 ):
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1: Derive kinematic equations for projectile motion
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
The ball is kicked from point A with the initial velocity of 10m/s. Determine the range R, and the
speed when the ball strikes the ground.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 3
Determine the speed at which the basketball at A must be thrown at the angle of 30° so that it
makes it to the basket at B.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 4
If a projectile is fired from point A in the direction shown and
with initial velocity of 150m/s, determine how long it is in the
air and the total distance it travels horizontally.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. The downward acceleration of an object in free flight is:
a) Zero
b) Increasing with time
c) 9.81 𝑚/𝑠 !
d) 9.81 𝑓𝑡/𝑠 !
2. The horizontal component of velocity remains _______ during a free-flight motion.
a) Zero
b) Constant
c) 9.81 𝑚/𝑠 !
d) 32.2 𝑓𝑡/𝑠 !
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.7 CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS
Objectives:
• Understand the normal and tangential coordinate system.
• Determine the normal and tangential components of velocity and acceleration of a particle
traveling along a curved path.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1
A car travels along the road shown to the right with a speed of 2𝑠 𝑚/𝑠
(s is in meters). If the radius of curvature of the road is 50 m,
determine the magnitude of the car’s acceleration when 𝑠 = 10𝑚.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
The boat shown starts from rest and travels around a circular path
with radius of 50 𝑚 at a speed which increases with time and
which can be defined by the relationship: 𝑣(𝑡) = 0.0625𝑡 ! 𝑚/𝑠.
At the instant when 𝑡 = 10𝑠, determine the magnitude of the
boat’s velocity and acceleration.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 3
If the train engine has a speed of 20 𝑚/𝑠 and an acceleration
of 14 𝑚/𝑠 ! acting in the direction shown at 𝐸, determine the
rate of increase in the train’s speed and the radius of curvature
of the path.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. If a particle moves along a curve with a constant speed, then its tangential component of
acceleration is:
a) Positive
b) Negative
c) Zero
d) Constant
2. The normal component of acceleration represents:
a) The time rate of change in the magnitude of the velocity
b) The time rate of change in the direction of the velocity
c) The magnitude of the velocity
d) The direction of the total acceleration
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS
Objectives:
• Understand the polar (2D) and cylindrical (3D) coordinate systems.
• Determine the velocity and acceleration components of a particle using cylindrical
components.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1
The platform shown is rotating such that, at any instant, its angular
position is determined by 𝜃(𝑡) = 4𝑡 (/! radians (t is in seconds).
The ball shown rolls outward so that its position is determined by
𝑟(𝑡) = 0.1𝑡 ( meters. Find the magnitude of the velocity and
acceleration of the ball when 𝑡 = 1.5𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
The arm of the robot is extending in the radial direction at a
constant rate of 1.5 𝑓𝑡/𝑠 when 𝑟 = 3𝑓𝑡, 𝑧 = 4𝑡 ! 𝑓𝑡 𝑎𝑛𝑑 𝜃 =
0.5𝑡 𝑟𝑎𝑑 (t is in seconds). Find the velocity and acceleration of
the grip 𝐴 when 𝑡 = 3 𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. In a polar coordinate system, the 𝜃̇ represents the:
a) Transverse velocity
b) Radial velocity
c) Angular velocity
d) Angular acceleration
2. The speed of a particle in a cylindrical coordinate system is:
a) 𝑟̇
b) 𝑟𝜃̇
!
c) QR𝑟𝜃̇S + 𝑟̇ !
!
d) QR𝑟𝜃̇S + 𝑟̇ ! + 𝑧̇ !
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
12.9 ABSOLUTE DEPENDENT MOTION ANALYSIS OF TWO PARTICLES
Objectives:
• Relate positions, velocities, and accelerations of particles undergoing dependent motion.
Consider a pulley system with two blocks 𝑨 and 𝑩:
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 1
Determine the velocity of block 𝐷 if the end of the cord at 𝐴 is
pulled down with a speed of 3𝑚/𝑠.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 2
Determine the time needed for the load at 𝐵 to attain a speed of
10𝑚/𝑠, starting from rest, if the cable is drawn into the motor
with an acceleration of 3𝑚/𝑠 ! .
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Ex. 3
If block 𝐴 of the pulley system is moving downward
at 6 𝑓𝑡/𝑠 while block 𝐶 is moving down at 18 𝑓𝑡/𝑠,
determine the relative velocity of block 𝐵.
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CVEN 363
Engineering Mechanics: Dynamics
Ch. 12 Kinematics of a Particle
Paal
Concept Inventory:
1. When particles are interconnected by a cable, the motions of the particles are:
a) Always independent
b) Always dependent
c) Not always dependent
d) None of the above
2. If the motion of one particle is dependent on that of another particle, each coordinate axis
system for the particles:
a) Should be directed along the path of motion
b) Can be directed anywhere
c) Should have the same origin
d) None of the above
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