Two dimensional motion, Slides of Physics

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4: Motion in 2 D

Steps to add Vectors

Graphically (qualitative)

Head to Tail method

Algebraic (quantitative)

set up a coordinate sys

draw a sketch of the vectors to be added

break up all vectors into x and y components

add all x components

add all y components

Position Vector in 2D

I am standing (5 miles, 7 miles). x and y coordinates

j

miles

i

miles

r

ˆ

7

ˆ

5

=

v

Recall,are the unit vectors.

j

and

i

ˆ

ˆ

Write this as:

Unit vectors givedirection. They do not changemagnitude.

Any vector can bewritten in terms of theunit vectors:

To find the magnitude ofvector A:

To find components:

j

A

i

A

A

y

x

v

A

v

x

A

y

A

(

)

(

)

2

2

y

x

A

A

A

From Pythagorean Theorem

cos

A

A

x

θ

sin

A

A

y

θ

45

sin

1

1

r

r

y

=

2-D displacement Vector

1 vr

Find x and y components of r

1

, r

2

.

2 vr

All in x direction.

in x and ydirection.

y

r

1 v

x

r

1 v

x

x

x

r

r

R

2

1

=

2 vr

x

r

1 v

x

R

y

y

y

r

r

R

2

1

=

y

r

2 v

y

R

y

r

1 v

45

o

45

cos

1

1

r

r

x

=

2

2

r

r

x

=

2

2

r

r

x

=

x

x

x

r

r

R

2

1

=

2 vr

x

r

1 v

x

R

45

cos

1 r

2 r

y

y

y

r

r

R

2

1

=

y

r

2 v

y

R

y

r

1 v

0

45

sin

1

r

m

m

sin

= + = m 1.

12

=

m

m

m

cos

x

R

y

R

m

1 .

7

=

Instantaneous Velocity

dt

r

d

t r

v

t

inst

v

v

v

=

Δ Δ

=

→

Δ

0

lim

1 vr

y

x

2 v r^ vr

Δ

v r

Zoom in

1 vr

' 2 vr

v r

becomes

tangent topath as

t

shrinks to

zero

inst

v

v

is tangentto path atany point

General Equations for 2-D motion

j

y

i

x

r

ˆ

ˆ

=

v

j

v

i

v

v

y

x

ˆ

ˆ

=

v

{

}

{

}

j t a t v y i t a t v x r

y

y

x

x

ˆ

ˆ

2

1 2

1

1

2

1 2

1

1

• • • • • = v

{

}

{

}

j t a v i t a v v

y

y

x

x

ˆ

ˆ

1

1

=

v

The motions in the x and y direction are completely

independent.

Clicker Question Sequence: 2D motion 1 (ignore air friction in these questions)

A ball is thrown into the air at an angle of 30

o

from the horizontal.

Consider the motion of the ball after leaving the hand and beforehitting the ground.When the ball reaches the highest point in its trajectory describethe velocity of the ball.

The velocity of the ball...

The magnitude of the ...

The magnitude of the ...

None of the above.

50%

19%

31%

0%

The velocity of the ball is zero.

The magnitude of the velocity is atits greatest value.

The magnitude of the velocity is atits smallest value (but greater thatzero).

None of the above.

Initial

response(s)

Correct response

A projectile is launched from a cannon at three different angles: 30

o

, 60

o

and 90

o

.

In each case the projectile is launched with the

same speed.At which angle does the projectile spend the most time in the air?

30o

60o

90o

In all launches, the p...

Not enough information.

20% 20%

20%

20%

20%

Initial

response(s)

Correctresponse

30

o

60

o

90

o

In all launches, the projectilespends the same amount oftime in the air.

Not enough information.

A. Both are hit simultan...

B.

0%

43%

57%

A battleship simultaneously fires two shells with the same initialspeed at enemy ships. If the shells follow the parabolictrajectories shown, which ship gets hit first?

A.

Both are hit simultaneously.

B.

Initial

response(s)

Correctresponse

Developed by OSU

What equation should we use for the motion in the xdirection?

What equation should we use for the motion in the ydirection?

t

v

x

x

=

Δ

t

a

v

v

y

y

y

=

1

2

1 2

1

t

a

t

v

y

y

y

=

Δ

x

x

v

v

1

=

Accelerated motion

Constant velocity

a

y

= g = -9.

m

/

s

2

A rock is launched with an initial speed

v

1

v

1

v

1x

g

v

1y

a

y

= g:

the

y

velocity decreases by 9.

m

s

each second

Δ

v

y

v

2

v

4x

v

4y

v

3

v

4

x

velocity is constant

y

velocity changes