Transcript PPT

Chapter 3
Kinematics in Two Dimensions
Vectors
3-1 Vectors & Scalars
•A vector has magnitude as well as direction.
•Examples: displacement, velocity, acceleration,
force, momentum
•A scalar has only magnitude
•Examples: time, mass, temperature
3-2 Vector Addition – One Dimension
A person walks 8 km
East and then 6 km East.
Displacement = 14 km
East
A person walks 8 km East
and then 6 km West.
Displacement = 2 km
3-2 Vector Addition
Example 1: A person walks 10 km
East and 5.0
 km North
DR  D1  D2
DR  D12  D22
DR  (10.0 km) 2  (5.0 km) 2  11.2 km
sin  
  sin 1 (
D2
DR
D2
5.0 km
)  sin 1 (
)  26.5 0
DR
11.2 km
Order doesn’t matter
3-2 Graphical Method of Vector Addition
Tail to Tip Method

V1

VR

V2

V3
3-2 Graphical Method of Vector Addition
Tail to Tip Method

V1

V2

V1

V3

VR

V2

V3
Parallelogram Method
3-3 Subtraction of Vectors
Negative of vector has same
magnitude but points in the
opposite direction.
For subtraction, we add the negative vector.
3-3 Multiplication by a Scalar
A vector V can be multiplied by a scalar c; the result is a
vector cV that has the same direction but a magnitude
cV. If c is negative, the resultant vector points in the
opposite direction.
Module 5 - 8
3-4 Adding Vectors by Components
Any vector can be expressed as the sum
of two other vectors, which are called its
components. Usually the other vectors are
chosen so that they are perpendicular to
each other.
Trigonometry Review
Hypotenuse
Opposite
Adjacent
Opposite
Hypotenuse
Adjacent
cos  
Hypotenuse
sin  
Opposite
sin 
tan  

Adjacent
cos 
3-4 Adding Vectors by Components
If the
components are
perpendicular,
they can be
found using
trigonometric
functions.
sin  
Opposite
Hypotenuse

Vy
V
Adjacent
Vx

Hypotenuse
V
Opp
sin 
tan  

Adj
cos 
cos  
 V y  V sin 
 Vx  V cos 
3-4 Adding Vectors by Components
The components are effectively one-dimensional,
so they can be added arithmetically:
Module 5 - 12
Signs of Components
y
Rx  
Rx  
Ry  
Ry  
x
Rx  
Ry  
Rx  
Ry  
3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
sin  
Vy
V
Example 2
A man pushing a mop across a floor causes it to undergo two
displacements. The first has a magnitude of 150 cm and makes a angle of 1200
with the positive x-axis. The resultant displacement has a magnitude of 140 cm
and is directed at an angle of 35.00 to the positive x axis. Find the magnitude
and direction of the second displacement.

A
A  150 cm
  
R  A B

B

R
120 0
35
R  140 cm
B R A
B R A
X
0
X
X
Bx  (140cm) cos 35  (150cm) cos 120
Bx  190cm
B R A
y
y
y
By  (140cm) sin 35  (150cm) sin 120
B y  49.6cm
Example 2
Alternative Solution. In the solution below, the angles
for vector A are measured from the negative x axis. In this case, we have to
assign the signs for the components. The answer is the same.

A

B

R
60
A  150 cm
  
R  A B
0
35
R  140 cm
B R A
B R A
X
0
X
X
Bx  (140cm) cos 35  (150cm) cos 60
Bx  190cm
B R A
y
y
y
By  (140cm) sin 35  (150cm) sin 60
B y  49.6cm
Example 2 Continued
A man pushing a mop across a floor
causes it to undergo two displacements. The first has a magnitude
of 150 cm and makes a angle of 1200 with the positive x-axis. The
resultant displacement has a magnitude of 140 cm and is directed
at an angle of 35.00 to the positive x axis. Find the magnitude and
direction of the second displacement.
B B B
2
2
x
y
B  (190cm) 2  (49.6cm) 2
sin  
 49.6cm
196cm
  49.6 
  sin 

196


1
 14.6 
 196 cm
3-5/3-6 Projectile Motion
A projectile is an object
moving in two
dimensions under the
influence of Earth's
gravity; its path is a
parabola.
Projectile Motion
•Neglect air resistance
•Consider motion only after release and before it hits
•Analyze the vertical and horizontal components separately
(Galileo)
•No acceleration in the horizontal, so velocity is constant
•Acceleration in the vertical is – 9.8 m/s2 due to gravity and
thus velocity is not constant.
•Object projected horizontally will reach the ground at the
same time as one dropped vertically
Equations for Projectile Motion
Horizontal
ax=0
Vertical
ay = - g
vx= constant
v0  vx 0
x  x0  vx0 t
vy  vy 0  g t
1 2
y  y0  v y 0 t  g t
2
v y2  v y20  2 g ( y  y0 )
Initial Velocity
v y 0  v0 sin 
vx 0  v0 cos 
•If the ball returns to the y = 0 point, then the velocity at
that point will equal the initial velocity.
•At the highest point, v0 y = 0 and v = vx0
Example 3A A football is kicked at an angle of 50.00 above the
horizontal with a velocity of 18.0 m / s. Calculate the maximum
height. Assume that the ball was kicked at ground level and lands
at ground level.
v  (18.0 m )(cos 50.0 )  11.6 m s
s
v  (18.0 m )(sin 50.0 )  13.8 m s
s

x0

y0
at top:
vy  vy0  g t  0
 v  13.8 m s
t   
 1.41s
m
 g  9.80 s
1
y  y  v t  gt
2
y0
2
2
max
0
yo
1
ymax  0  (13.8 m )(1.41s )   (9.8 m 2 )(1.41s ) 2
s
s
2
ymax  9.7m
Level Horizontal Range
•Range is determined by time it takes for ball to return to
ground level or perhaps some other vertical value.
•If ball hits something a fixed distance away, then time is
determined by x motion
•If the motion is on a level field, when it hits: y = 0
1
1
y  y0  v y 0 t  g t 2  0  0  v yo t  g t 2
2
2
2 v y0
Solving we find t  g
We can substitute this in the x equation to find the range R
R  x  vx 0 t  vxo (
2 v y0
g
)
2 vx 0 v yo
g
2 v02 sin  0 cos  0

g
Level Horizontal Range
We can use a trig identity 2 sin  cos   sin 2
v02 sin 2
R
g
•Greatest range:  = 450
•  = 300 and 600 have
same range.
(   450  150 )
Caution– the range formula
has limited usefulness. It is
only valid when the
projectile returns to the same
vertical position.
Example 3B
A football is kicked at an angle of 50.00 above the horizontal with
a velocity of 18.0 m / s. Calculate the range. Assume that the ball
was kicked at ground level and lands at ground level.
Assume time down = time up
For Range:
t  (2)(1.41s)  2.82s
R  x  x v t
 33m
0
x0
 0  (11.6 m )(2.82s)
s
Could also use range formula


v02 sin 2 (18 m / s ) 2 sin (2) ( 500 )
R

 33 m
2
g
9.8 m / s
Example 4A A football is kicked at an angle of 50.00 above the
horizontal with a velocity of 18.0 m / s. The football hits a window
in a house that is 25.0 m from where it was kicked. How high was
the window above the ground.
v  (18.0 m )(cos 50.0 )  11.6 m s
s
v  (18.0 m )(sin 50.0 )  13.8 m s
s

x0

y0
xv t
Time to hit the window:
x0
25.0m

11.6 m
s
x
t
v
x0
 2.16s
1
y  y  v t  gt
2
0
2
y0
1
m
y  0  (13.8
)( 2.16 s )  (9.8 m 2 )( 2.16s ) 2
s
s
2
y  6.9m
Example 4 B What is the final velocity and angle of the football
that hit the window in Example 4 A.
t  2.16s
v  v  gt
y
y0
v y  (13.8 m )  (9.8 m 2 ) (2.16s )
s
s
v  11.6 m
x
s
v  (11.6 m )  (7.37 m )
s
s
2
v
tan 
v
 7.37 m
2
 13.7 m
y
x
7.37 
  tan 

1
 11.6 
 32.4

below x axis
s
s
Example 5. (35) A rescue plane wants to drop supplies to isolated mountain
climbers on a rocky ridge 235 m below. If the plane is traveling horizontally
with a speed of 250 km /h (69.4 m / s) how far in advance of the recipients
(horizontal distance) must the goods be dropped (Fig. 3–37a)? .
vy0  0
v x 0  69.4 m / s
Coordinate system is 235
m below plane
1
y  235 m  0  g t 2  0
2
( 2 ) (235 m )
t
 6.93 s
2
9.8 m / s
x  x0  vxo t  0  ( 69.4 m / s ) ( 6.93 s )
x  481 m
3-7 Projectile Motion Is Parabolic
In order to demonstrate that
projectile motion is parabolic,
the book derives y as a function
of x. When we do, we find that it
has the form:
This is the
equation for
a parabola.
3-8 Relative Velocity
•Will consider how observations made in different
reference frames are related to each other.
A person walks toward the front of a train at 5 km / h
(VPT). The train is moving 80 km / h with respect to the
ground (VTG). What is the person’s velocity with respect
to the ground (VPG)?



VPG  VPT  VTG
VPG  5 km / h  80 km / h  85 km / h
3-8 Relative Velocity
•Boat is aimed upstream
so that it will move
directly across.
•Boat is aimed directly
across, so it will land at a
point downstream.
•Can expect similar
problems with airplanes.
Example 6 An airplane is capable of flying at 400 mi/h in still air.
At what angle should the pilot point the plane in order for it to
travel due east, if there is a wind of speed 50.0 mi/h directed due
south? What is the speed relative to the ground?
VPA


VAG

VPG
V AG
sin  
VPA
mi 

50
.
0

1 
0
1  VAG 
h


 sin 
 7.18
  sin 


400 mi
 VPA 
h  North of East

Module 6 - 6
VPG  ( 400 mi
h
) cos ( 7.18 0 )  397 mi
h