Download motion in one dimension

MOTION IN ONE
DIMENSION
• DYNAMICS – the study of motion and of
physical concepts (such as forces & mass)
• KYNEMATICS – the part of dynamics that
describe motion
MOTION IN ONE DIMENSION
• Any motion involve the concept of:
- displacement
- velocity
- acceleration
• In this chapter we use these concepts to
study the motion of objects undergoing
constant acceleration
DISPLACEMENT
• Motion involve the displacement of an
object from one place in space and time to
another
• The DISPLACEMENT Δx of an object is
defined as its change in position, and is
given by:
Δx = xf - xi
SI unit: m
(Δ- denote any change of quantity)
• VECTOR quantity - is characterized by
having magnitude (size) and a direction
• SCALAR quantity – is characterized by
having magnitude but not direction
• DISPLACEMENT – is a vector , because
has both, magnitude and direction
VELOCITY
• SPEED → Scalar → magnitude
• VELOCITY →vector → magnitude &
direction
• The average speed of an object over a
given time interval is defined as the total
distance traveled divided by the total time
elapsed:
v = d/t
SI unit: m/s
AVERAGE VELOCITY
• The average velocity v during a time
interval Δt is the displacement Δx divided
by Δt
v = Δx / Δt = (vf-vi) / ( tf -ti)
SI unit: m/s
• The velocity can be + or - ; depending on
the sign of the displacement
GRAPHICAL INTERPRETATION
OF VELOCITY
• Fig 1
• If a car move along the x-axis from A to B, the
graph – straight line, because the car is moving
with a ct. velocity, the same displacement Δx
occurs in each time Δt
• Fig 2 if is not a straight line, because velocity of
the car is changing
INSTANT VELOCITY
• The average velocity doesn’t take into
account the details of what happened
during an interval of time
• The instant velocity v – is the limit of the
average velocity as the time interval Δt
becomes infinitesimally small
v ≈lim Δt→ 0 (Δx / Δt)
SI unit: m/s
• The SLOPE of the
tangent line to the
position vs. time curve at
a “ given time” is defined
to be the instantaneous
velocity at that time
(fig 3)
• The instantaneous
speed of an object, which
is a scalar quantity, is
defined as the magnitude
of the instantaneous
velocity
ACCELERATION
• The velocity of a car increase harder when you
step harder on the gas pedal, and decrees when
you apply the brake
• The changing of an object’s velocity with time is
called acceleration
• The average acceleration a during the time int.
Δt is the change of velocity Δv divided by Δt
a = Δv/ Δt= (vf- vi)/ tf-ti)
SI unit: m/s2
• An average acceleration of 5 m/s 2 means on
average, the car increase its velocity by 5m/s
every second in the + direction
• When the object velocity and acceleration are in
the same direction, the speed of the object
increases with time
• When the object’s velocity and acceleration are
in opposite directions, the speed of the object
decreases in time
INSTANT ACCELERATION
• Fig 4
• The instantaneous
acceleration a is the limit of
the average acceleration as
the time interval Δt goes to
zero
a ≈lim Δt→0 (Δv/ Δt)
SI unit: m/s2
• The instantaneous
acceleration of an object at a
given time = the SLOPE of the
tangent to the velocity vs. time
graph at that time
MOTION DIAGRAM
• Velocity and acceleration are sometimes confused with
each other
• A motion diagram is a representative of a moving object
at successive intervals (fig5)
a) the car moves the same distance in each
interval. The car moves with ct. + velocity
and has zero acceleration
b) the velocity vector increases with time.
The car is moving with a + velocity and a
ct. +acceleration
c) The car slows. The car is moving with +
velocity but with a - acceleration
ONE DIMENSIONAL MOTION
WITH CT. ACCELERATION
• When an object moves with ct.
acceleration, the instantaneous
acceleration at any point in a time
interval is equal to the value of
the average acceleration over the
entire time interval (fig6)
• a=a
a =(vf -vi)/ tf -ti)
• If
ti =0; tf =t; vi =v0; vf =v
• a= (v- v0)/t or
• v = v0 + at
velocity as a function of time
• a steadily changes the initial
velocity v0 by an amount at (fig7)
• Because the velocity is increasing or
decreasing uniformly with time, we can
express the average velocity as the
arithmetic average
• v = (v0 + v)/2
• Δx= v t= [(v0 + v)/2] t
• Δx= ½ (v0 + v)t for a= ct.
• Δx= ½ (v0+v+at) t
• Δx= v0 t + ½ at2
displacement as a function of time (for
ct. a)
• The area under the graph
of v versus t for any object
is = to the displacement Δx
of the object
• Δx= ½(v+v0)(v-v0)/a= (v2v02)/2a
• v2= v02+ 2aΔx
velocity as a function of
displacement (for ct. a)
FREE FALLING OBJECT
• When air resistance is negligible, all objects
dropped under influence of gravity near Earth's
surface, fall towards Earth with the same ct.
acceleration (law discovered by Galileo Galilei)
• FREE FALLING object – any object moving
freely under the influence of gravity alone,
regardless of its initial motion
• gEarth =9.8 m/s2 (for quick estimates g≈10 m/s2 )