INTRODUCTION &
RECTILINEAR KINEMATICS: CONTINUOUS MOTION
Today’s Objectives:
Students will be able to:
1. Find the kinematic quantities
(position, displacement, velocity,
and acceleration) of a particle
traveling along a straight path.
An Overview of Mechanics
Mechanics: The study of how bodies
react to forces acting on them.
Statics: The study of
bodies in equilibrium.
Dynamics:
1. Kinematics – concerned with
the geometric aspects of motion
2. Kinetics - concerned with
the forces causing the motion
Kinematics
is the branch of classical mechanics that describes the motion of
points, bodies (objects) and systems of bodies (groups of
objects) without consideration of the causes of motion. The
study of kinematics is often referred to as the geometry of
motion.
To describe motion, kinematics studies the trajectories
of points, lines and other geometric objects and their
differential properties such as velocity and acceleration. In
mechanical engineering, robotics and biomechanics[8] to
describe the motion of systems composed of joined parts
(multi-link systems) such as an engine, a robotic arm or the
skeleton of the human body.
RECTILINEAR KINEMATICS: CONTINIOUS MOTION
(Section 12.2)
A particle travels along a straight-line path
defined by the coordinate axis s.
The position of the particle at any instant,
relative to the origin, O, is defined by the
position vector r, or the scalar s. Scalar s
can be positive or negative. Typical units
for r and s are meters (m) or feet (ft).
The displacement of the particle is
defined as its change in position.
Vector form:  r = r’ - r
Scalar form:  s = s’ - s
The total distance traveled by the particle, sT, is a positive scalar
that represents the total length of the path over which the particle
travels.
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.
The average velocity of a particle during a
time interval t is
vavg = r / t
The instantaneous velocity is the time-derivative of position.
v = dr / dt
Speed is the magnitude of velocity: v = ds /dt
It is how the position of a point changes with each
instant of time. speed is also non-negative.
Average speed is the total distance traveled divided by elapsed time:
(vsp)avg = sT / t
ACCELERATION
Acceleration is the rate of change in the velocity of a particle. It is a
vector quantity. Typical units are m/s2 or ft/s2.
The instantaneous acceleration is the time
derivative of velocity.
Vector form: a = dv / dt
Scalar form: a = dv / dt = d2s / dt2
Acceleration can be positive (speed
increasing) or negative (speed decreasing).
As the book indicates, the derivative equations for velocity and
acceleration can be manipulated to get a ds = v dv
SUMMARY OF KINEMATIC RELATIONS:
RECTILINEAR MOTION
• Differentiate position to get velocity and acceleration.
v = ds/dt ;
a = dv/dt or a = v dv/ds
• Integrate acceleration for velocity and position.
Position:
Velocity:
v
t
v
s
 dv =  a dt or  v dv =  a ds
s
t
 ds =  v dt
vo
o
vo
so
so
o
• Note that so and vo represent the initial position and
velocity of the particle at t = 0.
CONSTANT ACCELERATION
The three kinematic equations can be integrated for the special case
when acceleration is constant (a = ac) to obtain very useful equations.
A common example of constant acceleration is gravity; i.e., a body
freely falling toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2
downward. These equations are:
v
t
 dv =  a
dt
yields
v = vo + act
 ds =  v dt
yields
s = s o + v ot + (1/2) a c t 2
yields
v 2 = (vo )2 + 2ac(s - so)
vo
o
s
t
so
v
c
o
s
 v dv =  ac ds
vo
so
Procedure for Analysis
Coordinate System.
• Establish a position coordinate s along the path and specify its fixed origin and
positive direction.
• Since motion is along a straight line, the vector quantities position, velocity, and
acceleration can be represented as algebraic scalars. For analytical work the sense
of s, v, and a is then defined by their algebraic signs.
• The positive sense for each of these scalars can be indicated by an arrow shown
alongside each kinematic equation as it is applied.
Kinematic Equations.
• If a relation is known between any two of the four variables a, v, s and t, then a
third variable can be obtained by using one of the kinematic equations, a = dv/ dt,
v = ds/ dt or a ds = v dv, since each equation relates all three variables. *
• Whenever integration is performed, it is important that the position and velocity
be known at a given instant in order to evaluate either the constant of integration
if an indefinite integral is used, or the limits of integration if a definite integral is
used.
• Remember that Eqs. 12-4 through 12-6 have only limited use. These equations
apply only when the acceleration is constant and the initial conditions are s = So
and v = Vo when t = O.
EXAMPLE
Given: A particle travels along a straight line to the right
with a velocity of v = ( 4 t – 3 t2 ) m/s where t is
in seconds. Also, s = 0 when t = 0.
Find: The position and acceleration of the particle
when t = 4 s.
Plan: Establish the positive coordinate, s, in the direction the
particle is traveling. Since the velocity is given as a
function of time, take a derivative of it to calculate the
acceleration. Conversely, integrate the velocity
function to calculate the position.
Solution:
EXAMPLE
(continued)
1) Take a derivative of the velocity to determine the acceleration.
a = dv / dt = d(4 t – 3 t2) / dt =4 – 6 t
=> a = – 20 m/s2 (or in the  direction) when t = 4 s
2) Calculate the distance traveled in 4s by integrating the
velocity using so = 0:
s
t
v = ds / dt => ds = v dt =>  ds =  (4 t – 3 t2) dt
so
o
=> s – so = 2 t2 – t3
=> s – 0 = 2(4)2 – (4)3 => s = – 32 m ( or )
GROUP PROBLEM SOLVING
Given:Ball A is released from rest
at a height of 40 ft at the
same time that ball B is
thrown upward, 5 ft from the
ground. The balls pass one
another at a height of 20 ft.
Find:The speed at which ball B was
thrown upward.
Plan: Both balls experience a constant downward acceleration
of 32.2 ft/s2 due to gravity. Apply the formulas for
constant acceleration, with ac = -32.2 ft/s2.
Solution:
GROUP PROBLEM SOLVING
(continued)
1) First consider ball A. With the origin defined at the ground,
ball A is released from rest ((vA)o = 0) at a height of 40 ft
((sA )o = 40 ft). Calculate the time required for ball A to drop to
20 ft (sA = 20 ft) using a position equation.
sA = (sA )o + (vA)o t + (1/2) ac t2
So,
20 ft = 40 ft + (0)(t) + (1/2)(-32.2)(t2) => t = 1.115 s
Solution:
GROUP PROBLEM SOLVING
(continued)
2) Now consider ball B. It is throw upward from a height of 5 ft
((sB)o = 5 ft). It must reach a height of 20 ft (sB = 20 ft) at the
same time ball A reaches this height (t = 1.115 s). Apply the
position equation again to ball B using t = 1.115s.
sB = (sB)o + (vB)ot + (1/2) ac t2
So,
20 ft = 5 + (vB)o(1.115) + (1/2)(-32.2)(1.115)2
=> (vB)o = 31.4 ft/s