Chapter_4 MOTION IN TWO DIMENSIONAL AND THREE DIMENSIONAL.ppt

Chapter_4 MOTION IN TWO DIMENSIONAL AND THREE DIMENSIONAL.ppt

1. Chapter 4 Motion in Two and Three
Dimensions
4.1. What is Physics?
      
4.2. Position and Displacement
      
4.3. Average Velocity and Instantaneous Velocity
      
4.4. Average Acceleration and Instantaneous
Acceleration
      
4.5. Projectile Motion
      
4.6. Projectile Motion Analyzed
      
4.7. Uniform Circular Motion
      
4.8. Relative Motion in One Dimension
      
4.9. Relative Motion in Two Dimensions

2. What is Physics?

3. Position and Displacement
Position vector:
Displacement :

4. EXAMPLE 1:
 Displacement
1
r

2
r

In Fig., the position vector for a
particle is initially at
and then later is
What is the particle's displacement from

to
?

5. Problem 2
A rabbit runs across a parking lot on
which a set of coordinate axes has,
strangely enough, been drawn. The
coordinates of the rabbit’s position as
functions of time t (second) are given
by
At t=15 s, what is the
rabbit’s position vector in
unit-vector notation and in
magnitude-angle
notation?

6. Average and Instantaneous Velocity
Instantaneous velocity is:

7. Particle’s Path vs Velocity
Displacement: The velocity vector
The direction of the instantaneous velocity of
a particle is always tangent to the particle’s
path at the particle’s position.

8. Problem 3
A rabbit runs across a parking lot on
which a set of coordinate axes has,
strangely enough, been drawn. The
coordinates of the rabbit’s position as
functions of time t (second) are given
by
At t=15 s, what is the
rabbit’s velocity vector in
unit-vector notation and in
magnitude-angle
notation?

9. Average and Instantaneous Acceleration
Average acceleration is
Instantaneous acceleration is
yx z
x y z
vv vv
a i j k a i a j a k
t t t t
 
      
   

    


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 




 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

10. Speed up or slow down
•If the velocity and acceleration components
along a given axis have the same sign then they
are in the same direction. In this case, the object
will speed up.
•If the acceleration and velocity components have
opposite signs, then they are in opposite
directions. Under these conditions, the object will
slow down.

11. Problem 4
A rabbit runs across a parking lot on
which a set of coordinate axes has,
strangely enough, been drawn. The
coordinates of the rabbit’s position as
functions of time t (second) are given
by
At t=15 s, what is the rabbit’s
acceleration vector in unit-
vector notation and in
magnitude-angle notation?

12. How to solve two-dimensional motion problem?
One ball is released from rest at the same instant that
another ball is shot horizontally to the right
The horizontal and vertical
motions (at right angles to each
other) are independent, and the
path of such a motion can be
found by combining its horizontal
and vertical position components.
By Galileo

13. Projectile Motion
A particle moves in a
vertical plane with some
initial velocity but its
acceleration is always the
free-fall acceleration g,
which is downward. Such a
particle is called a
projectile and its motion is
called projectile motion.

14. Properties of Projectile Motion

The Horizontal Motion:
•no acceleration
•velocity v
x
remains
unchanged from its initial
value throughout the motion
•The horizontal range R is
maximum for a launch angle
of 45°
The vertical Motion:
•Constant acceleration g
•velocity v
y
=0 at the highest
point.
  
  



  
                                                                                                            

15. Check Your Understanding
A projectile is fired into the air, and it follows the
parabolic path shown in the drawing. There is no
air resistance. At any instant, the projectile has a
velocity v and an acceleration a. Which one or
more of the drawings could not represent the
directions for v and a at any point on the
trajectory?

16. Example
 4
 
 
A Falling Care Package
Figure shows an airplane moving
horizontally with a constant velocity
of +115 m/s at an altitude of 1050 m.
The directions to the right and
upward have been chosen as the
positive directions. The plane
releases a “care package” that falls
to the ground along a curved
trajectory. Ignoring air resistance,
(a). determine the time required for
the package to hit the ground.
(b) find the speed of package B and
the direction of the velocity vector
just before package B hits the
ground.

17. Example
 5
 
 
The Height of a Kickoff
A placekicker kicks a football at an angle of θ=40.0
o
above the
horizontal axis, as Figure shows. The initial speed of the ball
is
(a) Ignore air resistance and find the maximum height H that the
ball attains.
(b) Determine the time of flight between kickoff and landing.
(c). Calculate the range R of the projectile.

18. UNIFORM CIRCULAR MOTION
Uniform circular motion is the motion of an
object traveling at a constant (uniform) speed
on a circular path



  
                                                                                      

19. Properties of UNIFORM CIRCULAR MOTION
•Period of the motion T: is the time for a particle to go
around a closed path exactly once has a special name.
Average speed is :
This number of revolutions in a given time is known as
the frequency, f.

20. Example 6 
 
A Tire-Balancing Machine
The wheel of a car has a radius of r=0.29
m and is being rotated at 830 revolutions
per minute (rpm) on a tire-balancing
machine. Determine the speed (in m/s) at
which the outer edge of the wheel is
moving.

21. CENTRIPETAL ACCELERATION
•Magnitude: The centripetal acceleration
of an object moving with a speed v on a
circular path of radius r has a magnitude
ac given by
2
c
v
a
R

•Direction: The centripetal
acceleration vector always points
toward the center of the circle and
continually changes direction as
the object moves.

23. Check Your Understanding
The car in the drawing is
moving clockwise around a
circular section of road at a
constant speed. What are
the directions of its velocity
and acceleration at following
positions? Specify your
responses as north, east,
south, or west.

(a)position 1
(b) position 2

24. Example
 7
 
 
The Effect of Radius on Centripetal Acceleration
The bobsled track at the
1994 Olympics in
Lillehammer, Norway,
contained turns with radii
of 33 m and 24 m, as
Figure illustrates. Find the
centripetal acceleration at
each turn for a speed of
34 m/s, a speed that was
achieved in the two-man
event. Express the
answers as multiples of
g=9.8 m/s2.

25. Relative Motion in One Dimension
PA PB BA
v v v 
xPA xPB xBA
a a a 

  
                                                                                                           
The coordinate   
                
The velocity   
                
The acceleration   
                
PA PB BA
x x x 

26. Relative Motion in Two Dimension
PA PB BA
v v v 
  
PA PB BA
a a a 
  

  
                                                                                                           
The coordinate   
                
The velocity   
                
The acceleration   
                
PA PB BA
r r r 
  

27. Sample Problem
In Fig. 4-23a, a plane moves due
east while the pilot points the
plane somewhat south of east,
toward a steady wind that blows
to the northeast. The plane has
velocity relative to the
wind, with an airspeed (speed
relative to the wind) of 215 km/h,
directed at angle θ south of
east. The wind has velocity v
pG

relative to the ground with speed
of 65.0 km/h, directed 20.0° east
of north. What is the magnitude
of the velocity of the plane
relative to the ground, and what
is θ?