2-D Kinematics Review

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14 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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2
-
D Kinematics Review

Scalars
vs

Vectors


Scalars

have magnitude only


Distance, speed, time, mass


Vectors

have both magnitude and direction



displacement, velocity, acceleration

R



head

tail

Inverse Vectors


Inverse vectors have the same length, but opposite
direction.

A

-
A

A

B

R

A

+
B

=
R

Graphical Addition of Vectors


Vectors are added graphically together head
-
to
-
tail.


The sum is called the resultant.


The inverse of the sum is called the equilibrant

Unit Vectors


Unit vectors are quantities that specify direction only.
They have a magnitude of exactly one, and typically
point in the x, y, or z directions.


ˆ
points in the x direction
ˆ
points in the y direction
ˆ
points in the z direction
i
j
k
Unit Vectors

z

y

x

i

j

k

Unit Vectors


Instead of using magnitudes and directions, vectors can be
represented by their components combined with their unit
vectors.


Example: displacement of 30 meters in the +x direction added to a
displacement of 60 meters in the

y direction added to a
displacement of 40 meters in the +z direction yields a displacement
of:

ˆ
ˆ ˆ
(30 -60 40 ) m
30,-60,40 m
i j k
 
 

Adding Vectors Using Unit Vectors


Simply add all the
i

components together, all
the
j

components together, and all the
k

components together.

Sample problem:
Consider two vectors,
A

= 3.00
i

+ 7.50
j

and
B

=
-
5.20
i

+ 2.40

j.
Calculate
C

where
C

=
A

+
B
.

Sample problem:
Consider two vectors,
A

= 3.00
i

+ 7.50
j

and
B

=
-
5.20
i

+ 2.40

j.
Calculate
C

where
C

=
A

+
B
.

Suppose I need to convert unit vectors to
a magnitude and direction?


Given the vector

2 2 2
ˆ
ˆ ˆ
x y z
x y z
r r i r j r k
r r r r
  
  
Sample problem:
You move 10 meters north and 6 meters east. You
then climb a 3 meter platform, and move 1 meter west on the
platform. How far are you from your starting point?

Sample problem:
You move 10 meters north and 6 meters east. You
then climb a 3 meter platform, and move 1 meter west on the
platform. How far are you from your starting point?

1 Dimension

2 or 3 Dimensions


x: position



x: displacement


v: velocity


a: acceleration


r
: position



r
: displacement


v
: velocity


a
: acceleration


r

= x
i

+ y
j

+ z
k



r

=

x
i

+

y
j

+

z
k


v

= v
x

i

+ v
y

j

+ v
z

k


a

= a
x

i

+ a
y

j

+ a
z

k

In Unit Vector

Notation

Sample problem:

The position of a particle is given by


r

= (80 + 2t)
i



40
j

-

5t
2
k.
Derive the velocity and acceleration
vectors for this particle. What does motion “look like”?


Sample problem:

The position of a particle is given by


r

= (80 + 2t)
i



40
j

-

5t
2
k.
Derive the velocity and acceleration
vectors for this particle.

Trajectory of Projectile

g

g

g

g

g


This shows the parabolic trajectory of a projectile fired over
level ground.


Acceleration points down at 9.8 m/s
2

for the entire
trajectory.

Trajectory of Projectile

v
x

v
y

v
y

v
x

v
x

v
y

v
x

v
y

v
x


The velocity can be resolved into components all along its path.
Horizontal velocity remains constant; vertical velocity is
accelerated.

Remember…

To work projectile problems…


…resolve the initial velocity into components.



V
o

V
o,y
= V
o

sin


V
o,x
= V
o

cos


Sample problem:

A soccer player kicks a ball at 15 m/s at an angle
of 35
o

above the horizontal over level ground. How far
horizontally
will the ball travel until it strikes the ground?

Sample problem:

A soccer player kicks a ball at 15 m/s at an angle
of 35
o

above the horizontal over level ground. How far will the ball
travel until it strikes the ground?

Sample problem:

A cannon is fired at a 15
o

angle above the
horizontal from the top of a 120 m high cliff. How long will it take
the cannonball to strike the plane below the cliff? How far from the
base of the cliff will it strike?

Sample problem:

A cannon is fired at a 15
o

angle above the
horizontal from the top of a 120 m high cliff. How long will it take
the cannonball to strike the plane below the cliff? How far from the
base of the cliff will it strike?

Uniform Circular Motion


Occurs when an object moves in a circle without
changing speed.


Despite the constant speed, the object’s
velocity

vector
is continually changing; therefore, the object must be
accelerating.


The acceleration vector is pointed toward the center of
the circle in which the object is moving, and is referred
to as
centripetal acceleration
.

Vectors in

Uniform Circular Motion

a

v

a = v
2

/ r

v

a

v

a

v

a

Sample Problem

The Moon revolves around the Earth every 27.3 days. The radius
of the orbit is 382,000,000 m. What is the magnitude and
direction of the acceleration of the Moon relative to Earth?

Sample Problem

The Moon revolves around the Earth every 27.3 days. The
radius of the orbit is 382,000,000 m. What is the magnitude
and direction of the acceleration of the Moon relative to
Earth?

Tangential acceleration


Sometimes the speed of an object in circular motion is not
constant (in other words, it’s not uniform circular motion).


An acceleration component may be tangent to the path, aligned
with the velocity. This is called
tangential acceleration
. It causes
speeding up or slowing down.


The centripetal acceleration component causes the object to
continue to turn as the tangential component causes the speed
to change. The centripetal component is sometimes called the
radial acceleration
, since it lies along the radius.

v

Tangential Acceleration

radial or centripetal
component (a
r

or a
c
)

tangential component (a
T
)

a

If tangential acceleration
exists, either the speed or
the radius must change.
This is no longer UCM.

Sample Problem:

Given the figure at right rotating at
constant radius, find the radial and tangential
acceleration components if


= 30
o

and
a

has a
magnitude of 15.0 m/s
2
. What is the speed of the
particle at the location shown? How is the particle’s
speed changing?

5.00 m

a



v

Sample Problem:

Given the figure at right rotating at
constant radius, find the radial and tangential
acceleration components if


= 30
o

and
a

has a
magnitude of 15.0 m/s
2
. What is the speed of the
particle? How is it behaving?

5.00 m

a