# Kinematics In Two Dimensions

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```					Kinematics in Two Dimensions                                                                                       01/09/2008 03:00 PM

Kinematics in Two Dimensions

Kinematics in 2-dimensions. By the end of
this you will

1.    Remember your Trigonometry
2.    Know how to handle vectors
3.    be able to handle problems in 2-dimensions
4.    understand projectile motion

Swish! by alanfreed

Two (or Three) Dimensional Motion

For example: Projectiles, Planets, Pendulum, ....

Need to have some new mathematical techniques to do this: however you may need to revise your
basic trigonometry

Basic Trigonometry

Basic Definitions of trig relationships

A useful mnemomic

SohCahToa
θ =      a
R
Sin = opposite = d
hypotenuse  R
Cos = adjacent = R-δ
hypotenuse  R
Tan = opposite = d

Basic Trig.

Useful relationships

sin2θ + cos2θ = 1 (all values of θ)
tan θ = sin θ / cos θ

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

These are not so useful

cot θ = 1/tan θ
sec θ = 1/cos θ
cosec θ = 1/sin θ

Special Values

sin(450) = sin(π/4) = 1/√2
cos(450) = cos(π/4) = 1/√2
tan(45 0) = tan(π/4) = 1
sin(300) = sin(π/6) = 1/2
cos(600) = cos(π/3) = 1/2
sin(600) = cos(300) = √3/2
tan(30 0) = tan(π/3) = 1/√3
tan(60 0) = tan(π/6) = √3
sin(900) = sin(π/2) = 1
cos(900) = cos(π/2) = 0
tan(90 0) = tan(π/2) = ∞

These are worth writing down, but they are very easy to work out.

We often have a convenient simplification
when the angles are small
θ = a/R ~ sin(θ) = d/R ~ tan(θ) =                                d (R-δ)
cos(θ) ~ 1

Note that this only works if we measure angles in

2π radians = 360 0

Vectors

Need new mathematics to describe three dimensional objects:

Scalars: quantities with only magnitude

Vectors have direction as well

Scalars                 Vectors
Temperature Force

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

Speed                   Velocity
Density                 Acceleration
Time?                   Time?

Need a new symbol: can use

~ which is obvious
a
a which gets confused with other symbols
a, which is hard to read

We will use both ~ and a,
a

Need to be able to describe vectors in terms of scalar quantities: can do this in terms of
components: the projection of the vector along each axis

These are the components of the vector ~
a

Note that the components of a vector are scalars

Also can do this in terms of length of the vector and angle(s).

These two descriptions are related
a x = a cos(θ)
a y = a sin(θ)

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

Adding vectors: put them nose to
tail. Easy diagramatically

If we want to add them
algebraically, we just add the
components:

~ = ~ +~
c a b

means

cx = a x + b x
cy = a y + b y

To subtract vectors, Flip the vector round: the negative of a vector must add to the
vector to give zero

~ - ~ = ~ + ( -a) = 0
a a a          ~

Note that this means that the negative of a vector just has all its components reversed

The length of a vector is given by Pythagoras: 3-D analog of Pythagoras:

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Kinematics in Two Dimensions                                                                             01/09/2008 03:00 PM

Write this as
q
j~j =
r         x 2 + y2 + z 2

Note:

The length of a vector is a scalar.
Displacement is a vector.
The length of the displacement vector is the distance

Examples

e.g. a boat sails 10 km North -East and 5
km South: how must it sail to get to its
start?

How do we write this as a vector sum?

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

N.E means "equal components along the x and y directions" so first
step is

e.g. Motion by a car.

A car travels 5 km N, 10 km E, and then 15 km S. The components of vector ~ that describes this
a
are

1 . a x = -10
a y = 20

2 . a x = 10
a y = -10

3 . a x = 10
a y = 10

4 . a x = -10
a y = -10

We write this as

Projectile Motion

(can usually be treated as 2-D)

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

Treat position, r,
velocity v and
acceleration as 2-D
vectors. In
general,motion in one
direction can be
treated independently
of motion in a
second.

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

We can also see this
quantitatively

In general there are three independent vector
quantities:

The position r, the velocity v and the acceleration a.
However we have to treat the components
separately.

It is easiest to treat the two motions as independent 1-D motions

vy = v0y + ay t
vx = v0x + ax t                 1
1       y = v0y t + 2 ay t2
x = v0x t + 2 ax t2
a = À g(usually)
ax = 0(usually) y
= À 9:8ms À2

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM
= À 9:8ms
Can combine these to give (e.g) an equation for the range R:

What is value of t when the height y = 0?

2v0 sin ( Ò )
t=
g

(or t = 0: what does this mean?)
Hence Range
2
v0 sin ( 2Ò )
R=
g

e.g. a ball is thrown at 15 ms-1, at 300 to the horizontal.

How far does it go
How long is it in the air?

Relative Velocity
Example: a woman who can swim at 2 m/s is swimming
How fast    in a river which flows at .7 m/s.
does she
swim
upstream?
How fast
does she
swim
downstream?

How about a round trip?

A woman swims 100 m upstream at 2 m/s in a river with a current of .7 m/s, and then 100 m
downstream to return to her starting point. Compared to swimming 200 m in still water, does her
journey

1 . Take longer?
2 . Take a shorter time?
3 . Take exactly the same time?

As a somewhat
more
sophisticated
example: a
woman who can

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Kinematics in Two Dimensions                                                                       01/09/2008 03:00 PM

woman who can
swim at 2 m/s is
swimming in a
river which flows
at .7 m/s.

At what angle
should she swim to
reach a point on the
opposite bank
immediately
opposite the point
from which she
starts?

Now we want to describe why things move , not just how.

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